Import Libraries

rm(list=ls())
library(plotly)
## Loading required package: ggplot2
## 
## Attaching package: 'plotly'
## The following object is masked from 'package:ggplot2':
## 
##     last_plot
## The following object is masked from 'package:stats':
## 
##     filter
## The following object is masked from 'package:graphics':
## 
##     layout
library(ROCR)
library(readr); library(dplyr); library(tidyr); library(ggplot2)
## 
## Attaching package: 'dplyr'
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
library(rmarkdown)
library(GGally)
## Registered S3 method overwritten by 'GGally':
##   method from   
##   +.gg   ggplot2
library(tree)
library(DataExplorer)
library(factoextra)
## Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBa
library(randomForest)
## randomForest 4.7-1.2
## Type rfNews() to see new features/changes/bug fixes.
## 
## Attaching package: 'randomForest'
## The following object is masked from 'package:dplyr':
## 
##     combine
## The following object is masked from 'package:ggplot2':
## 
##     margin
library(clValid)
## Loading required package: cluster

Data Description

Load Data

The data set is unstructured and we want to import data into a tidy format (i.e. dataframe)

# To input this unstructured file into R we use read_delim 
Customers <- read_delim("/Users/ogheneatoma/Documents/ST309 final project - SmartRetail/data/marketing_campaign.csv", delim = "\t")
## Rows: 2240 Columns: 29
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: "\t"
## chr  (3): Education, Marital_Status, Dt_Customer
## dbl (26): ID, Year_Birth, Income, Kidhome, Teenhome, Recency, MntWines, MntF...
## 
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
# To show # of observations and features in this data set
dim(Customers)
## [1] 2240   29

Now we have a tidy data set with each feature forming a column and each value having its own cell

Basic examination of the dataset

Column names:

names(Customers)
##  [1] "ID"                  "Year_Birth"          "Education"          
##  [4] "Marital_Status"      "Income"              "Kidhome"            
##  [7] "Teenhome"            "Dt_Customer"         "Recency"            
## [10] "MntWines"            "MntFruits"           "MntMeatProducts"    
## [13] "MntFishProducts"     "MntSweetProducts"    "MntGoldProds"       
## [16] "NumDealsPurchases"   "NumWebPurchases"     "NumCatalogPurchases"
## [19] "NumStorePurchases"   "NumWebVisitsMonth"   "AcceptedCmp3"       
## [22] "AcceptedCmp4"        "AcceptedCmp5"        "AcceptedCmp1"       
## [25] "AcceptedCmp2"        "Complain"            "Z_CostContact"      
## [28] "Z_Revenue"           "Response"

Unique values in each feature

nunique <- function(x) length(unique(x))
nunique_counts <- sapply(Customers, nunique)
nunique_counts
##                  ID          Year_Birth           Education      Marital_Status 
##                2240                  59                   5                   8 
##              Income             Kidhome            Teenhome         Dt_Customer 
##                1975                   3                   3                 663 
##             Recency            MntWines           MntFruits     MntMeatProducts 
##                 100                 776                 158                 558 
##     MntFishProducts    MntSweetProducts        MntGoldProds   NumDealsPurchases 
##                 182                 177                 213                  15 
##     NumWebPurchases NumCatalogPurchases   NumStorePurchases   NumWebVisitsMonth 
##                  15                  14                  14                  16 
##        AcceptedCmp3        AcceptedCmp4        AcceptedCmp5        AcceptedCmp1 
##                   2                   2                   2                   2 
##        AcceptedCmp2            Complain       Z_CostContact           Z_Revenue 
##                   2                   2                   1                   1 
##            Response 
##                   2

Summary statistics

summary(Customers)
##        ID          Year_Birth    Education         Marital_Status    
##  Min.   :    0   Min.   :1893   Length:2240        Length:2240       
##  1st Qu.: 2828   1st Qu.:1959   Class :character   Class :character  
##  Median : 5458   Median :1970   Mode  :character   Mode  :character  
##  Mean   : 5592   Mean   :1969                                        
##  3rd Qu.: 8428   3rd Qu.:1977                                        
##  Max.   :11191   Max.   :1996                                        
##                                                                      
##      Income          Kidhome          Teenhome      Dt_Customer       
##  Min.   :  1730   Min.   :0.0000   Min.   :0.0000   Length:2240       
##  1st Qu.: 35303   1st Qu.:0.0000   1st Qu.:0.0000   Class :character  
##  Median : 51382   Median :0.0000   Median :0.0000   Mode  :character  
##  Mean   : 52247   Mean   :0.4442   Mean   :0.5062                     
##  3rd Qu.: 68522   3rd Qu.:1.0000   3rd Qu.:1.0000                     
##  Max.   :666666   Max.   :2.0000   Max.   :2.0000                     
##  NA's   :24                                                           
##     Recency         MntWines         MntFruits     MntMeatProducts
##  Min.   : 0.00   Min.   :   0.00   Min.   :  0.0   Min.   :   0   
##  1st Qu.:24.00   1st Qu.:  23.75   1st Qu.:  1.0   1st Qu.:  16   
##  Median :49.00   Median : 173.50   Median :  8.0   Median :  67   
##  Mean   :49.11   Mean   : 303.94   Mean   : 26.3   Mean   : 167   
##  3rd Qu.:74.00   3rd Qu.: 504.25   3rd Qu.: 33.0   3rd Qu.: 232   
##  Max.   :99.00   Max.   :1493.00   Max.   :199.0   Max.   :1725   
##                                                                   
##  MntFishProducts  MntSweetProducts  MntGoldProds    NumDealsPurchases
##  Min.   :  0.00   Min.   :  0.00   Min.   :  0.00   Min.   : 0.000   
##  1st Qu.:  3.00   1st Qu.:  1.00   1st Qu.:  9.00   1st Qu.: 1.000   
##  Median : 12.00   Median :  8.00   Median : 24.00   Median : 2.000   
##  Mean   : 37.53   Mean   : 27.06   Mean   : 44.02   Mean   : 2.325   
##  3rd Qu.: 50.00   3rd Qu.: 33.00   3rd Qu.: 56.00   3rd Qu.: 3.000   
##  Max.   :259.00   Max.   :263.00   Max.   :362.00   Max.   :15.000   
##                                                                      
##  NumWebPurchases  NumCatalogPurchases NumStorePurchases NumWebVisitsMonth
##  Min.   : 0.000   Min.   : 0.000      Min.   : 0.00     Min.   : 0.000   
##  1st Qu.: 2.000   1st Qu.: 0.000      1st Qu.: 3.00     1st Qu.: 3.000   
##  Median : 4.000   Median : 2.000      Median : 5.00     Median : 6.000   
##  Mean   : 4.085   Mean   : 2.662      Mean   : 5.79     Mean   : 5.317   
##  3rd Qu.: 6.000   3rd Qu.: 4.000      3rd Qu.: 8.00     3rd Qu.: 7.000   
##  Max.   :27.000   Max.   :28.000      Max.   :13.00     Max.   :20.000   
##                                                                          
##   AcceptedCmp3      AcceptedCmp4      AcceptedCmp5      AcceptedCmp1    
##  Min.   :0.00000   Min.   :0.00000   Min.   :0.00000   Min.   :0.00000  
##  1st Qu.:0.00000   1st Qu.:0.00000   1st Qu.:0.00000   1st Qu.:0.00000  
##  Median :0.00000   Median :0.00000   Median :0.00000   Median :0.00000  
##  Mean   :0.07277   Mean   :0.07455   Mean   :0.07277   Mean   :0.06429  
##  3rd Qu.:0.00000   3rd Qu.:0.00000   3rd Qu.:0.00000   3rd Qu.:0.00000  
##  Max.   :1.00000   Max.   :1.00000   Max.   :1.00000   Max.   :1.00000  
##                                                                         
##   AcceptedCmp2        Complain        Z_CostContact   Z_Revenue 
##  Min.   :0.00000   Min.   :0.000000   Min.   :3     Min.   :11  
##  1st Qu.:0.00000   1st Qu.:0.000000   1st Qu.:3     1st Qu.:11  
##  Median :0.00000   Median :0.000000   Median :3     Median :11  
##  Mean   :0.01339   Mean   :0.009375   Mean   :3     Mean   :11  
##  3rd Qu.:0.00000   3rd Qu.:0.000000   3rd Qu.:3     3rd Qu.:11  
##  Max.   :1.00000   Max.   :1.000000   Max.   :3     Max.   :11  
##                                                                 
##     Response     
##  Min.   :0.0000  
##  1st Qu.:0.0000  
##  Median :0.0000  
##  Mean   :0.1491  
##  3rd Qu.:0.0000  
##  Max.   :1.0000  
## 

Data Cleaning

Missing Values

plot_missing(Customers)

Customers <- na.omit(Customers)

There are 24 observations dropped in the data set because of missing values.

Duplicate Values

Check if there are duplicated data on the same customer characterised by their ID.

duplicates <- Customers$ID[duplicated(Customers$ID)]
duplicates
## numeric(0)

No duplicates found.

Outliers

Based on the outliers found in the summary statistics, we will examine

par(mfrow = c(1, 2))
hist(2021-Customers$Year_Birth)
hist(Customers$Income)

Based on the histogram plots, we can see the majority of individuals are aged below 80 and have an income level below 100,000. Hence we drop the unusual values.

Removing the outliers

Customers_cleaned <- subset(Customers, 2021-Customers$Year_Birth < 80 & Income < 100000)

This drops another 24 observations.

Feature Engineering/Data Transformation

# Age category for each customer assuming the data was collected in 2021
Customers_cleaned$Age <- 2021 - Customers_cleaned$Year_Birth

# Cut into differnet age groups by generations for our analysis  
Customers_cleaned$AgeCategory <- cut(Customers_cleaned$Age, c(0, 40, 56, 78), c('<40', '41-56', '>57')) 

# Number of Children
Customers_cleaned$NumChildren <- Customers_cleaned$Kidhome + Customers_cleaned$Teenhome

# Total spending
Customers_cleaned$Spending <- Customers_cleaned$MntWines + Customers_cleaned$MntFruits + Customers_cleaned$MntMeatProducts + Customers_cleaned$MntFishProducts + Customers_cleaned$MntSweetProducts + Customers_cleaned$MntGoldProds

# Log transformation of highly-skewed spending variables
Customers_cleaned <- Customers_cleaned %>%
  mutate(
    log_Wines = log(1+MntWines),
    log_Fruits = log(1+MntFruits),
    log_MeatProducts = log(1+MntMeatProducts),
    log_FishProducts = log(1+MntFishProducts),
    log_SweetProducts = log(1+MntSweetProducts),
    log_GoldProds = log(1+MntGoldProds),
    log_Spending = log(1+Spending) 
  )

# Relationship
Customers_cleaned$Relationship <- ifelse(Customers_cleaned$Marital_Status %in% c("Married", "Together"), 1, 0)
Customers_cleaned$Relationship <- factor(Customers_cleaned$Relationship, levels = c(0, 1), labels = c("Not Partnered", "Partnered"))

# Education
Education <- c(Basic = "Bachelors", '2n Cycle' = "Bachelors", Graduation = "Graduate", Master = "Masters", PhD = "PhD")
Customers_cleaned$Education <- as.character(Education[Customers_cleaned$Education])
Customers_cleaned$Education <- factor(Customers_cleaned$Education)

# Number of years customers joined 
Dt_Customer <- as.Date(Customers_cleaned$Dt_Customer, format = "%d-%m-%Y")
Year_Customer <- as.numeric(format(Dt_Customer, "%Y"))
Customers_cleaned$YearsJoined <- 2021 - Year_Customer

# Number of accepted campaigns out of 6 in total 
Customers_cleaned$TotalAcceptedCmp <- Customers_cleaned$AcceptedCmp1 + Customers_cleaned$AcceptedCmp2 + Customers_cleaned$AcceptedCmp3 + Customers_cleaned$AcceptedCmp4 + Customers_cleaned$AcceptedCmp5 + Customers_cleaned$Response

# Remove redundant columns
Customers_cleaned <- subset(Customers_cleaned, select = -c(ID, Z_CostContact, Z_Revenue, Year_Birth, Marital_Status, Dt_Customer, Teenhome, Kidhome))

Convert all variables into numerical using label encoding.

# Examine data types of the columns 
str(Customers_cleaned)
## tibble [2,198 × 35] (S3: tbl_df/tbl/data.frame)
##  $ Education          : Factor w/ 4 levels "Bachelors","Graduate",..: 2 2 2 2 4 3 2 4 4 4 ...
##  $ Income             : num [1:2198] 58138 46344 71613 26646 58293 ...
##  $ Recency            : num [1:2198] 58 38 26 26 94 16 34 32 19 68 ...
##  $ MntWines           : num [1:2198] 635 11 426 11 173 520 235 76 14 28 ...
##  $ MntFruits          : num [1:2198] 88 1 49 4 43 42 65 10 0 0 ...
##  $ MntMeatProducts    : num [1:2198] 546 6 127 20 118 98 164 56 24 6 ...
##  $ MntFishProducts    : num [1:2198] 172 2 111 10 46 0 50 3 3 1 ...
##  $ MntSweetProducts   : num [1:2198] 88 1 21 3 27 42 49 1 3 1 ...
##  $ MntGoldProds       : num [1:2198] 88 6 42 5 15 14 27 23 2 13 ...
##  $ NumDealsPurchases  : num [1:2198] 3 2 1 2 5 2 4 2 1 1 ...
##  $ NumWebPurchases    : num [1:2198] 8 1 8 2 5 6 7 4 3 1 ...
##  $ NumCatalogPurchases: num [1:2198] 10 1 2 0 3 4 3 0 0 0 ...
##  $ NumStorePurchases  : num [1:2198] 4 2 10 4 6 10 7 4 2 0 ...
##  $ NumWebVisitsMonth  : num [1:2198] 7 5 4 6 5 6 6 8 9 20 ...
##  $ AcceptedCmp3       : num [1:2198] 0 0 0 0 0 0 0 0 0 1 ...
##  $ AcceptedCmp4       : num [1:2198] 0 0 0 0 0 0 0 0 0 0 ...
##  $ AcceptedCmp5       : num [1:2198] 0 0 0 0 0 0 0 0 0 0 ...
##  $ AcceptedCmp1       : num [1:2198] 0 0 0 0 0 0 0 0 0 0 ...
##  $ AcceptedCmp2       : num [1:2198] 0 0 0 0 0 0 0 0 0 0 ...
##  $ Complain           : num [1:2198] 0 0 0 0 0 0 0 0 0 0 ...
##  $ Response           : num [1:2198] 1 0 0 0 0 0 0 0 1 0 ...
##  $ Age                : num [1:2198] 64 67 56 37 40 54 50 36 47 71 ...
##  $ AgeCategory        : Factor w/ 3 levels "<40","41-56",..: 3 3 2 1 1 2 2 1 2 3 ...
##  $ NumChildren        : num [1:2198] 0 2 0 1 1 1 1 1 1 2 ...
##  $ Spending           : num [1:2198] 1617 27 776 53 422 ...
##  $ log_Wines          : num [1:2198] 6.46 2.48 6.06 2.48 5.16 ...
##  $ log_Fruits         : num [1:2198] 4.489 0.693 3.912 1.609 3.784 ...
##  $ log_MeatProducts   : num [1:2198] 6.3 1.95 4.85 3.04 4.78 ...
##  $ log_FishProducts   : num [1:2198] 5.15 1.1 4.72 2.4 3.85 ...
##  $ log_SweetProducts  : num [1:2198] 4.489 0.693 3.091 1.386 3.332 ...
##  $ log_GoldProds      : num [1:2198] 4.49 1.95 3.76 1.79 2.77 ...
##  $ log_Spending       : num [1:2198] 7.39 3.33 6.66 3.99 6.05 ...
##  $ Relationship       : Factor w/ 2 levels "Not Partnered",..: 1 1 2 2 2 2 1 2 2 2 ...
##  $ YearsJoined        : num [1:2198] 9 7 8 7 7 8 9 8 8 7 ...
##  $ TotalAcceptedCmp   : num [1:2198] 1 0 0 0 0 0 0 0 1 1 ...
##  - attr(*, "na.action")= 'omit' Named int [1:24] 11 28 44 49 59 72 91 92 93 129 ...
##   ..- attr(*, "names")= chr [1:24] "11" "28" "44" "49" ...
# Identify categorical columns
categorical_cols <- sapply(Customers_cleaned, is.factor)

# Apply label encoding to categorical columns
Customers_cleaned[categorical_cols] <- lapply(Customers_cleaned[categorical_cols], as.numeric)

Subset the data frame used for PCA

Customers_PCA <- subset(Customers_cleaned, select = -c(Spending,MntWines,MntFruits,MntMeatProducts,MntFishProducts,MntSweetProducts,MntGoldProds,AcceptedCmp1,AcceptedCmp2,AcceptedCmp3,AcceptedCmp4,AcceptedCmp5,Complain,Response, AgeCategory))

Data Analysis

EDA

Pick out on some key features for correlation analysis.

plot_correlation(Customers_cleaned)

plot_histogram(Customers_cleaned)

PCA

Now the columns of the data set contain the following variables.

names(Customers_PCA)
##  [1] "Education"           "Income"              "Recency"            
##  [4] "NumDealsPurchases"   "NumWebPurchases"     "NumCatalogPurchases"
##  [7] "NumStorePurchases"   "NumWebVisitsMonth"   "Age"                
## [10] "NumChildren"         "log_Wines"           "log_Fruits"         
## [13] "log_MeatProducts"    "log_FishProducts"    "log_SweetProducts"  
## [16] "log_GoldProds"       "log_Spending"        "Relationship"       
## [19] "YearsJoined"         "TotalAcceptedCmp"

We examine the mean and variances of different variables

apply(Customers_PCA, 2, mean)
##           Education              Income             Recency   NumDealsPurchases 
##        2.480437e+00        5.148159e+04        4.900773e+01        2.324841e+00 
##     NumWebPurchases NumCatalogPurchases   NumStorePurchases   NumWebVisitsMonth 
##        4.084167e+00        2.633303e+00        5.815742e+00        5.345314e+00 
##                 Age         NumChildren           log_Wines          log_Fruits 
##        5.207188e+01        9.517743e-01        4.683589e+00        2.244250e+00 
##    log_MeatProducts    log_FishProducts   log_SweetProducts       log_GoldProds 
##        4.130463e+00        2.540630e+00        2.244045e+00        3.126917e+00 
##        log_Spending        Relationship         YearsJoined    TotalAcceptedCmp 
##        5.623144e+00        1.645587e+00        7.972247e+00        4.440400e-01
apply(Customers_PCA, 2, var)
##           Education              Income             Recency   NumDealsPurchases 
##        9.088115e-01        4.230185e+08        8.364920e+02        3.554877e+00 
##     NumWebPurchases NumCatalogPurchases   NumStorePurchases   NumWebVisitsMonth 
##        7.255999e+00        7.765336e+00        1.044714e+01        5.816526e+00 
##                 Age         NumChildren           log_Wines          log_Fruits 
##        1.363344e+02        5.602586e-01        3.244145e+00        2.458667e+00 
##    log_MeatProducts    log_FishProducts   log_SweetProducts       log_GoldProds 
##        2.413958e+00        2.747391e+00        2.524857e+00        1.644474e+00 
##        log_Spending        Relationship         YearsJoined    TotalAcceptedCmp 
##        2.149672e+00        2.289086e-01        4.712367e-01        7.777046e-01

PCA with standardised variables

PCA <- prcomp(x = Customers_PCA, scale = TRUE)
Table_PCA <- rbind(PCA$rotation, summary(PCA)$importance)
knitr::kable(Table_PCA, digits = 4, align = 'c')
PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 PC9 PC10 PC11 PC12 PC13 PC14 PC15 PC16 PC17 PC18 PC19 PC20
Education 0.0099 0.2065 -0.5445 -0.1559 0.1085 -0.2216 -0.1056 0.4948 0.0411 0.4625 0.1228 -0.2081 0.1508 0.0390 0.0357 -0.0467 0.1170 0.0324 -0.0546 -0.0120
Income 0.3058 -0.0130 -0.2373 0.0273 -0.0125 0.0062 0.0785 -0.0277 -0.1706 -0.2592 -0.0195 -0.1899 -0.1028 -0.0239 0.0144 0.0783 -0.3221 0.5816 -0.4992 -0.0278
Recency 0.0068 0.0058 0.0537 0.4368 0.6357 -0.5579 0.2359 -0.1628 0.0379 0.0114 -0.0464 0.0174 0.0558 -0.0055 0.0316 0.0091 0.0166 0.0137 -0.0016 0.0010
NumDealsPurchases -0.0104 0.5535 0.1884 0.1225 -0.0611 0.1792 0.2604 0.0198 -0.2750 0.3564 -0.0283 0.4273 0.1076 -0.0857 -0.0196 0.3061 0.0176 0.1707 -0.0974 0.0317
NumWebPurchases 0.2254 0.3456 0.0570 -0.0440 -0.0460 -0.0024 0.1143 0.0210 0.5452 -0.1734 -0.2162 -0.1865 -0.3156 0.0654 -0.0310 0.2502 0.4719 0.0143 -0.0894 0.0416
NumCatalogPurchases 0.2883 -0.0400 -0.0817 -0.0631 0.0310 -0.0747 0.0025 -0.0931 -0.2272 0.1560 0.1442 0.3138 -0.5472 0.0117 -0.2987 -0.4736 0.2741 0.0679 0.0183 0.0312
NumStorePurchases 0.2760 0.0952 -0.0613 0.1198 -0.0464 0.0541 0.0156 0.2280 -0.0446 -0.4049 -0.2024 0.2737 0.5321 -0.0032 0.1097 -0.4168 0.2877 -0.0280 -0.0678 0.0202
NumWebVisitsMonth -0.2080 0.3490 0.2718 -0.2125 0.0135 -0.1036 -0.0231 -0.0196 0.3465 0.1250 -0.2421 -0.0155 -0.0225 -0.0219 -0.0248 -0.5513 -0.4183 0.1254 -0.1139 -0.0180
Age 0.0560 0.2103 -0.3554 0.2673 0.0829 0.2143 -0.5862 -0.5450 0.1302 0.1267 -0.1079 0.0727 0.0925 0.0206 -0.0019 -0.0034 -0.0075 -0.0333 -0.0373 0.0008
NumChildren -0.1881 0.3819 -0.0579 0.2206 -0.0329 0.1862 0.2114 -0.1085 -0.4217 -0.1310 0.0262 -0.5757 -0.1119 0.0523 0.0593 -0.2670 0.1006 -0.1977 0.0750 -0.0507
log_Wines 0.3035 0.2296 -0.1814 -0.0297 0.0016 -0.0413 0.0576 0.0507 0.0247 -0.1875 0.0289 0.0269 -0.0594 -0.0208 0.0129 0.0618 -0.3677 0.0669 0.6619 0.4367
log_Fruits 0.2709 -0.1363 0.2068 0.0901 0.0083 0.0890 -0.0546 0.0178 -0.0075 0.2371 -0.1709 -0.3440 0.1942 -0.6599 -0.3819 -0.0271 0.0736 0.0763 0.1005 0.0318
log_MeatProducts 0.3327 0.0417 0.0017 0.0007 0.0068 -0.0278 0.0307 0.0760 -0.0446 0.0424 -0.0396 0.0287 -0.0987 -0.0675 -0.0013 0.0606 -0.3094 -0.7196 -0.4329 0.2344
log_FishProducts 0.2695 -0.1488 0.2141 0.1147 -0.0118 0.0889 -0.0895 0.0444 -0.0358 0.3071 -0.1171 -0.0792 -0.1842 -0.0560 0.8020 -0.1110 0.0568 0.1016 0.0718 0.0370
log_SweetProducts 0.2682 -0.1394 0.2127 0.1034 -0.0303 0.0589 -0.0170 0.0436 -0.0954 0.2646 -0.2998 -0.1845 0.1568 0.7232 -0.2875 0.0194 -0.0603 0.0527 0.0725 0.0283
log_GoldProds 0.2452 0.0960 0.2511 -0.0175 0.0071 0.0739 0.0515 -0.1863 0.2328 0.0600 0.7900 -0.1388 0.2740 0.1196 0.0185 -0.1113 0.0374 0.0506 -0.0916 0.0936
log_Spending 0.3395 0.1317 -0.0207 -0.0244 -0.0030 -0.0243 0.0374 0.0231 0.0425 -0.0388 0.0567 0.0498 -0.0278 -0.0213 0.0079 0.0554 -0.2085 -0.1506 0.2244 -0.8567
Relationship -0.0087 0.0305 -0.0121 0.3209 -0.7386 -0.5750 -0.0405 -0.1074 0.0073 0.0603 0.0311 -0.0156 0.0279 -0.0284 0.0040 -0.0051 -0.0005 0.0026 -0.0029 0.0038
YearsJoined 0.0454 0.2580 0.3823 -0.2454 0.1398 -0.3168 -0.6075 0.1194 -0.3588 -0.2213 0.0528 -0.0787 -0.0273 0.0113 0.0038 0.1559 0.0934 0.0287 -0.0160 0.0105
TotalAcceptedCmp 0.1269 -0.0099 -0.1227 -0.6218 -0.0252 -0.2209 0.2584 -0.5326 -0.1693 0.0969 -0.1893 -0.0648 0.2478 -0.0095 0.1363 0.0368 0.1395 -0.0509 0.0187 0.0194
Standard deviation 2.8356 1.4195 1.2694 1.0745 1.0039 0.9764 0.9098 0.8908 0.7654 0.7058 0.6740 0.6399 0.5985 0.5555 0.5454 0.4990 0.4822 0.3183 0.2892 0.1316
Proportion of Variance 0.4020 0.1007 0.0806 0.0577 0.0504 0.0477 0.0414 0.0397 0.0293 0.0249 0.0227 0.0205 0.0179 0.0154 0.0149 0.0124 0.0116 0.0051 0.0042 0.0009
Cumulative Proportion 0.4020 0.5028 0.5834 0.6411 0.6915 0.7391 0.7805 0.8202 0.8495 0.8744 0.8971 0.9176 0.9355 0.9509 0.9658 0.9783 0.9899 0.9950 0.9991 1.0000
par(mfrow=c(1,1))
plot(Table_PCA['Proportion of Variance',], type = 'o', lwd = 5, col = 'blue', main = 'PC proportions of total variance', xlab = 'PC', ylab = 'Proportion of variance', axes = FALSE)
axis(1, 1:22)
axis(2)

Subset data based on chosen principle components

nf <- 3
Customers_clust <- as.data.frame(PCA$x[, 1:nf])
Customers_clust
##                PC1           PC2          PC3
## 1     3.9652558507  0.6607660795  1.591087289
## 2    -3.4722319455 -0.5637275979 -1.280295584
## 3     2.7525441587 -0.5272958013  0.346571407
## 4    -2.7696208652 -1.3743700228  0.358376316
## 5     0.9804712402  0.3991332033 -0.358985060
## 6     1.3043261953  0.7752718737 -0.695340025
## 7     1.7635014298  0.9867400903  1.789659077
## 8    -1.6775500664  0.5132352056 -0.046602800
## 9    -3.3100178001  0.0092184102 -0.927981220
## 10   -5.2371345187  2.1713880533 -0.630846939
## 11   -2.7583928842 -1.2670179871  2.639190479
## 12    3.0727704667 -1.7675497202  0.389638943
## 13   -0.6628823181  1.7770652767 -1.160078604
## 14   -3.1233231557 -1.4913145552  2.000037785
## 15    4.3490029545  0.4297254486 -1.743673033
## 16   -2.4418583812  0.8571300131  1.337978396
## 17    0.1459414193  0.1781602009  0.948443732
## 18    3.2807940346  1.9827950135 -1.033565728
## 19   -1.3813658331 -1.2117437717  3.112212179
## 20    0.0861094993 -0.5663084787  2.151368758
## 21    0.3948656122  1.8400223737  0.507799695
## 22   -0.2380085608  2.5147646708 -1.974358636
## 23    0.7506349783  0.7904782225 -2.606283209
## 24   -0.0400413525  2.6691918008  1.031036764
## 25   -2.0793564504 -1.2896012972  1.295254019
## 26   -1.0689270439  1.4161170214  0.626544647
## 27   -1.2204921389 -1.4192669914  0.802735769
## 28    3.8320679761 -0.8615875584 -2.189351919
## 29   -3.2703833477 -1.8318789285 -0.040975723
## 30    0.7149510517 -1.4444126008 -0.033144327
## 31   -2.5362109581  0.1216694939 -0.268080246
## 32    0.2639791612  2.5550996165 -0.061173080
## 33    2.7759716973 -0.2836910516 -0.671063804
## 34   -3.0298300683  0.5323407943 -1.568173770
## 35    1.9512811569  1.4928263035 -1.646633103
## 36   -2.3616482565 -0.5448546901  2.127933991
## 37   -3.1067010992 -1.0000362343 -0.903058769
## 38    3.0366948807  0.5024741609 -1.315797012
## 39    3.5176639598 -0.2477920599  1.128202946
## 40   -3.4311141848 -0.1808012065  0.408315071
## 41   -3.8438867784 -0.7304714598 -0.058717800
## 42   -3.6090983316 -0.4405422702  0.263005781
## 43    3.3168335607 -2.6323457754  0.326287835
## 44   -1.8283387565 -0.9710529817  3.989211939
## 45   -2.7970650449 -0.3751827533  0.883062861
## 46    2.3199647051  4.2715463838  0.319473409
## 47    1.6994362337 -1.2208635346 -0.993822941
## 48    4.4979383887 -0.2173494688 -0.293133029
## 49   -2.5654134179 -0.8756450957  0.799293114
## 50    4.7962525906 -2.2585565999 -0.327759179
## 51    1.3023736277  1.7083868510  0.848313769
## 52    5.5294531310 -0.7422988021 -0.564586793
## 53    3.5742952709 -1.3039524227  0.241890829
## 54   -3.4675245542 -2.1046530642 -0.440502698
## 55   -0.7574797751  1.8736213376 -2.010978198
## 56    3.9284316382 -1.4812804902 -0.890546150
## 57    1.1613315592  1.0828610617  0.411106976
## 58    1.9430305829  1.9063147534  0.463937456
## 59    0.1029854272  1.2582772758 -2.818562120
## 60    2.9267132792 -0.6690004453  0.032050570
## 61   -3.2898821931 -1.2778443413  0.774527052
## 62   -5.3049942725 -0.7696054380 -1.353404816
## 63    4.3204903088 -0.3937320821 -0.083791812
## 64    2.1321285874 -0.1667907722  0.059107453
## 65    1.9493603831  2.1129303688  0.951018857
## 66    3.2150680577 -0.1169948105 -0.182887381
## 67    3.1168597406  1.6718619363  1.089078859
## 68    0.2722197198  3.0737577545 -0.945189855
## 69   -1.5415584479 -2.2350892518 -0.816528730
## 70   -4.1005463925 -1.1939571037  0.683086167
## 71    3.4672052838 -0.1674536762  0.294530419
## 72    4.6710489370 -1.1824525131  0.914396606
## 73   -3.8032017196 -2.5601412470  0.608395229
## 74    0.2985519516  1.6002732453 -0.654841405
## 75   -1.5335110931  0.7197432338  2.065867208
## 76   -4.2985570264 -1.0132464757  0.046364114
## 77   -2.5645026047 -0.0948997355  1.538538623
## 78    0.7149510517 -1.4444126008 -0.033144327
## 79    3.7073531223 -2.4271159647 -0.295169061
## 80   -2.8814013558 -0.0809374860  0.280798966
## 81   -2.8073961246  1.9759029336 -0.925877777
## 82   -0.2125333784  1.1994083583 -1.110535999
## 83    3.9888325987 -1.8923549644  0.054106400
## 84   -0.9666111741 -1.3506620426 -1.408350732
## 85   -3.5230685385 -1.0314564334 -1.172501864
## 86   -3.5619491751 -0.1141971477 -2.065962887
## 87   -2.9115198410 -1.7914938980  0.432485923
## 88   -0.2184611183  2.6220984515  0.097134421
## 89   -3.3217907419  0.2104560599 -0.397116953
## 90    3.4943581404 -0.7449525808 -0.318276869
## 91    1.1958287650 -0.7837369260  0.153094497
## 92   -2.8453463206 -0.1177074887  0.313069401
## 93   -3.1937742254 -0.1343205436 -2.162555348
## 94    3.9785206616  0.0759014664  1.383159346
## 95    2.9459476733 -0.0417911254 -1.054892748
## 96    3.7278268377 -2.1677911018 -1.279785478
## 97   -3.6299149037 -1.8375179393  1.341828645
## 98   -1.9800754070  1.5501458393 -0.317809383
## 99    2.5144881176  1.2396861196  1.348237705
## 100  -1.6006871168  0.1649056599  1.283658605
## 101   2.7632430131 -0.7336668036 -0.249508904
## 102   3.2665328169 -0.5877542262  0.473173948
## 103   2.6700828420  0.6013339036  0.738083362
## 104  -0.3670268129  1.7086521052 -1.889731798
## 105   3.4761069715 -1.4352659426  0.060937159
## 106  -0.4904133352  2.7655692723 -0.259865613
## 107  -2.3547512902  0.5408185017 -0.495907591
## 108   4.1222850381 -2.5421166383 -0.876024021
## 109   1.0979349262  1.0240563978  3.564964851
## 110  -2.5219242660 -0.4107839665  0.954382074
## 111  -3.3371827471 -1.1514004129  0.769039916
## 112   2.1451019118  1.0065526156 -0.077088803
## 113  -2.2561771054 -0.3849636826 -1.263255473
## 114  -4.0176397646 -0.1571722016  0.327151581
## 115  -3.3322778244 -1.0753349790  0.335755918
## 116   2.3797426127 -1.5797385704  0.917150833
## 117   3.5362847327 -1.7887947460  0.327957442
## 118  -3.0057138303  0.6293489413 -2.436838952
## 119   1.4382019094  1.5337263522 -2.215585317
## 120   1.6441636057  0.6889222964  0.766183334
## 121   0.0253696598  3.5204700813 -0.488007154
## 122   2.6944603803 -1.1090516375  0.125104110
## 123  -1.7958720798  2.8591966940 -1.071377860
## 124   2.7006178135 -0.7157587048 -2.131963312
## 125  -4.0618226731 -0.9608663264  2.565677154
## 126  -2.9763093280  2.0661494793 -1.285996507
## 127  -2.9324795256 -0.3322271912 -1.796261601
## 128  -3.4165423452  0.5689530638 -1.172016700
## 129   4.0784568572 -0.9895688269  0.422384119
## 130   1.1746004209  0.6015153156 -0.586947287
## 131   3.9737959275 -0.9543866291 -0.179411112
## 132   2.2818062962  1.1803550102 -0.429980446
## 133  -0.1179119742 -0.3855062806  2.204727670
## 134   0.9382327607  1.2875400421  1.610124738
## 135  -2.4321280385 -0.9431313519  1.497916514
## 136  -4.0552091495 -0.9594306302  0.378099088
## 137  -1.7414615045  2.9738334655  2.346251288
## 138  -0.6162592639 -2.0791388398 -0.999270200
## 139   0.8930438021 -0.1977835646 -1.775555645
## 140   0.7757600129  0.6920155949 -0.063764297
## 141   0.4287781900  1.2977573418  2.417264174
## 142  -1.9093338872 -0.9568516760  1.237849585
## 143   0.9684153127  2.6390515242 -1.930000954
## 144   3.3209569738  0.2132125984 -0.886809551
## 145  -0.5909894468  0.8162827002 -0.203732153
## 146  -4.1726674745 -0.7920528403 -0.872310755
## 147  -0.3579880768 -1.7435022247  0.105974556
## 148   3.4867547622 -1.4181245283 -0.393747588
## 149  -5.0226742219  0.5218709224 -1.260618478
## 150   3.3259832577  1.8124611784 -1.044649135
## 151  -2.3297664739 -0.0375841735  0.762149843
## 152   2.4464508082 -0.6316359332 -1.692925980
## 153  -2.5020862933  0.9943284167  0.300740300
## 154   2.1215367148 -1.2024143750 -0.284846667
## 155  -0.5753197159 -0.9634341009 -1.160433377
## 156   3.2767512218 -2.3620560576  1.345606563
## 157  -2.0339434607 -0.2681048543  0.038629901
## 158  -3.5037710468 -0.5160380834 -0.252288397
## 159  -2.7638575173 -0.7072509780  1.484792575
## 160  -1.7290807210 -0.3025237911 -0.566312362
## 161  -3.0336766973  0.5656765295 -2.952222761
## 162  -2.0687860172 -0.9215876780 -1.287952848
## 163   4.0260788633  1.1059503573 -0.075157626
## 164   4.6859105458  0.5022075197 -1.029454895
## 165  -3.8973142900  0.4599550682 -2.111521209
## 166  -4.6686882460 -1.1126314299 -1.755788396
## 167   3.4943581404 -0.7449525808 -0.318276869
## 168  -1.3475272866 -0.4349155637 -1.774383538
## 169  -2.7543853965 -1.4475533379  1.588275231
## 170   3.3696008563 -0.1029516327 -1.206057267
## 171  -3.3507601479 -0.9868991025 -1.015662939
## 172  -2.7704369342 -2.2869429481  0.708948897
## 173  -3.1677564692 -0.5983518326  1.693300507
## 174  -3.9036341629 -0.5361973340 -2.267217736
## 175   0.8762051640  0.8285550249  0.067409608
## 176   3.8652564840 -1.6593986579 -0.023690609
## 177  -3.0697531900  0.8661691201 -1.630985427
## 178  -2.7101614693 -1.2727964731  2.152018855
## 179   4.1285265136 -0.6137671561  0.134799647
## 180  -2.1237709382 -0.4695014769  1.527021506
## 181  -0.6136743525  1.2391646980  0.407291993
## 182  -3.9539294821 -0.0401647682  0.415809059
## 183   2.2085514439 -1.0074318943 -1.088857817
## 184   3.5781838719 -1.1264866712 -0.500212836
## 185   3.7415778051 -1.1455748228 -0.063675779
## 186  -1.0789945701  2.2919410914 -0.880672285
## 187   0.7039770325  0.9423306914 -1.200893505
## 188   3.6786906724 -1.7229238679  0.443811374
## 189   3.2002412419  0.9680757429  1.297290249
## 190  -3.8930979155  0.6999328325 -1.671344042
## 191  -3.0201074062 -1.6663540038  0.812074252
## 192  -2.8217172182 -1.1321253405  1.512781445
## 193  -0.0453037123  1.6470339743  0.380133521
## 194  -3.3167412283 -0.7206107820 -1.244958762
## 195   3.5823761834  2.9467907415  1.287755739
## 196  -0.1787307356  0.3616264148  0.190394831
## 197   3.1263691886 -1.6929677091 -1.478968556
## 198  -2.1230829091  4.1067254336 -0.423205036
## 199  -0.1630088796  3.9759948599  0.637544292
## 200   3.0887191041 -2.0534428030  0.747503464
## 201   0.4096798652  1.5378526643  1.398915552
## 202  -4.9429283408 -0.4711171237 -0.496374512
## 203   4.0930884560 -0.1452542326 -1.772725613
## 204  -1.2094556450  1.1268149563 -0.967550776
## 205   2.9307777616  1.2757881788  1.467995834
## 206  -1.4250737868 -2.3560162581  0.744935817
## 207   2.4877927894 -1.7222833212 -0.413724664
## 208   1.1024533527  0.8887591842 -2.673322444
## 209  -2.4896362944  1.3957844810 -0.802583299
## 210   2.9024348541  2.4494977086  1.003435788
## 211  -2.0313602438  0.9667268673  0.648755746
## 212   0.2250672387  1.5400363083  1.557902713
## 213   0.5712673475  1.9178066254  0.141166137
## 214   3.4960568403 -0.8179890014 -0.536330953
## 215  -4.5165264447 -1.2375724332 -1.280274789
## 216  -0.0444893781  3.1089027943  1.025643900
## 217   2.7477707609 -0.3735219668  1.039364073
## 218  -3.8041033292 -1.0748187850  0.416035870
## 219  -3.6760575904 -2.5209269373 -0.846152492
## 220   2.8082459059  1.3176599572 -1.016281905
## 221  -2.3564219884 -1.1041864798  1.768495451
## 222  -4.5026749752 -0.7048844433 -0.593799254
## 223   1.7063546635  0.1682042383 -2.047792132
## 224  -2.7996327272 -0.0920344034  1.600314317
## 225   4.2321746359 -1.1500531719  0.329236340
## 226   2.6897284103  0.0233739055  1.102959324
## 227  -4.1187611118 -0.6191424066  0.345801397
## 228   4.3534573460 -1.9382626323  0.675319934
## 229   1.5866400117  3.9178196860  0.799612634
## 230  -4.6317499686  0.0316170136 -2.429680562
## 231   2.4277530530  0.5174455324 -0.394109486
## 232   2.9160195968  0.8386991532  0.222869915
## 233   4.1550623247 -0.8675825177  0.725320642
## 234   3.1188321083  1.7321387578  1.193664811
## 235  -2.7343516708 -0.1689000751  0.296808149
## 236  -3.7008377204 -1.1513988867  0.856447630
## 237  -0.1822427364 -0.7003299903  2.169782103
## 238   2.3681177998  2.2128511938  1.278882175
## 239  -3.9309976486  0.1919668929 -0.974881945
## 240   1.5958715714  1.9732982999 -0.633066961
## 241   0.6551597038  0.1717245083  1.069044352
## 242  -3.1023667424  0.2073952435  0.747308508
## 243  -4.8460102401  0.9646253464 -2.031684899
## 244  -3.3644535448 -0.7812331414  1.544980702
## 245   1.4779686587  2.3036673766 -1.277902366
## 246  -0.5316587620  0.2336451341 -0.694131550
## 247  -3.8881441314  0.0189901043 -0.095719436
## 248  -1.0984735682  2.8768633507 -0.729934110
## 249  -2.2793552818  1.2518020181 -0.104310623
## 250   4.3237403333 -2.0704633107 -0.374767743
## 251  -1.7191420023  2.1927897453 -0.383285516
## 252   3.0870063787 -2.0552518000  0.699846559
## 253  -2.3950122314 -0.7261074308  0.750844542
## 254   3.0639820197  0.0910143404  1.128873429
## 255  -2.5247769419 -0.7581208787 -0.434982697
## 256  -1.0239811830  0.0333686602 -2.067019897
## 257  -2.9776227192  0.1493457989 -0.277265561
## 258  -2.2366818055  0.5147723985  0.393608249
## 259   1.5701048953  0.8284179767 -1.674624152
## 260   3.5766229988 -0.3193519552  0.936571224
## 261   2.2468887085 -2.6204431160 -0.549625231
## 262   2.6195851902 -1.4709983746 -1.181757758
## 263   1.9656284111  0.9486870018 -0.113828942
## 264  -1.9688589003 -0.9520899295  2.543016364
## 265   0.2250672387  1.5400363083  1.557902713
## 266   0.1459414193  0.1781602009  0.948443732
## 267   1.2146643497 -0.5925564104 -1.551965760
## 268  -3.8111942214 -0.1768763651 -1.741142097
## 269  -2.2994463731 -0.7438722120  0.907654271
## 270   3.7373501964 -0.7875206747  0.084429751
## 271   2.2471386157  4.3934121729  1.257881239
## 272   4.1456423356 -2.0516665795 -0.212954898
## 273  -0.1803730149  1.3685778527 -0.310435306
## 274  -2.6939659424 -1.8503667026  1.292022699
## 275  -1.6630077335  0.1405420536  0.661940635
## 276   4.1859432056 -1.1373902306  1.296716042
## 277  -3.2147108794  0.2821961933  0.730797947
## 278  -4.0586585052 -0.5594121133 -0.230160539
## 279   1.1140054748  2.2755677363 -1.074112023
## 280  -3.2982887822 -1.7021231981  1.569731613
## 281  -2.4661358234 -0.2143565583  0.095597834
## 282   2.1646032712 -2.1465665036  0.422738020
## 283   0.6301749540  2.4435849557 -2.565979045
## 284  -2.8659789652  0.5389787589 -0.852204125
## 285   1.1462530280  0.8524842206  0.123213252
## 286  -2.4771984448 -1.1676455279 -2.200585143
## 287  -0.4423954052 -0.2379908771  1.362940678
## 288  -2.7150057872 -0.1104810504  1.220485160
## 289   4.3128816749 -1.1283570144 -0.419204842
## 290  -1.6121753460 -1.0718765160  2.120874769
## 291   2.8343810306 -1.6291841835 -2.024477976
## 292  -0.5347042389  1.4096329938 -1.696594888
## 293  -0.4999573882 -1.2816102626  0.072548570
## 294  -2.8454032481 -0.6708648987  0.963936979
## 295  -4.3445165243  0.8168717587 -0.349659185
## 296   3.4661883846 -0.9533136973 -0.336889080
## 297  -1.3619785823  1.7023839840 -0.386409072
## 298  -4.1376980879  0.1113437564 -2.140829664
## 299  -2.4047740205  0.4746727637  0.444526533
## 300   1.2173278805 -0.6254026483  1.411126612
## 301  -3.6960736974  0.0237016290 -0.714513271
## 302   1.9661886804 -0.1421659022  0.073814120
## 303  -3.4454583459 -1.2406639968  0.743889507
## 304  -4.5546866224 -0.1959387196 -2.603345057
## 305   2.4397389036  2.3791178698 -0.055399127
## 306   0.9228854601  2.4483260748  0.908327333
## 307   2.4976673081 -0.6830084610 -0.412593802
## 308  -2.9622924445  0.4106421134 -0.159763449
## 309  -3.8389916451 -0.9700386715  0.155833147
## 310  -4.0102177031  0.7240430254 -3.204407841
## 311  -1.5667813236  0.6471751247  0.291683876
## 312  -4.1731205181 -1.0705774230  0.736155095
## 313  -2.3208013838 -1.6144215306 -0.753648863
## 314   1.0009408705  1.9088981038 -2.228954456
## 315  -2.5949342469 -0.5345156182 -0.630888057
## 316  -3.6948015037 -0.4868788021  0.930583584
## 317   2.9846156023  0.2592005190  0.821209102
## 318   4.3124516454 -0.1387893041  0.625820887
## 319  -2.7571980707 -0.4173039586  1.327589701
## 320   4.1718111418 -0.6143247189 -0.243686422
## 321   3.4617071991 -1.2059709897  1.508197718
## 322  -1.0106874804 -1.7653117562  1.108231032
## 323   0.0509662855  2.5487011933 -1.346185274
## 324   1.2183822381  0.2527309653 -1.894095219
## 325  -4.0665001640 -1.3653890068  0.741847296
## 326   3.5991966205 -1.9763471712 -0.032993103
## 327  -3.7922535721  0.2199149167 -0.754034163
## 328   3.8032016692 -2.3431125511  0.003656751
## 329  -3.2993823020  0.6631578075  1.002082459
## 330  -4.3039812064 -1.2053360680 -1.345650685
## 331   2.3316470628  0.7873379302 -0.313293910
## 332   3.6076561599 -0.1095651065  1.112323884
## 333   1.8573060929 -0.4382234506 -1.937467851
## 334   2.6934081124 -1.1535247592  0.177783174
## 335   0.3085576711  3.3494845030 -0.880803494
## 336  -3.3560355322  0.1002678500  0.293092643
## 337  -2.4942897295  0.0135690898 -1.207480487
## 338   3.4779261007 -0.9321837119 -0.328311382
## 339   1.4750808750 -0.3814689545 -1.926199824
## 340   0.5691920147  1.0907242176 -1.386950495
## 341   4.9186813868 -0.9838303579  0.609535697
## 342   0.0648187392  1.2706069253 -2.554339091
## 343  -2.5825389721 -0.7557667812  0.040423182
## 344  -3.8881441314  0.0189901043 -0.095719436
## 345   0.3568362760  3.0656546914  1.501440394
## 346   1.3388802833  0.7534516013  0.899786329
## 347   3.5296700167 -1.0136665889 -0.848470040
## 348  -2.8028089978 -0.9731359441  1.174709278
## 349   3.7715252724 -0.6479724294  1.104050201
## 350   0.4742783623  1.2808602107 -0.279661785
## 351  -3.5227260658 -1.9909948734  0.498615349
## 352  -1.1555284000  2.5756466740  1.599407151
## 353  -3.1344970710 -1.2367592266 -0.679640320
## 354   2.1464351158  1.8376138915  1.053607502
## 355  -2.6731444898  0.9969442751 -0.933290525
## 356  -2.6173497869  2.0134942298  0.366605315
## 357  -2.2177888305 -1.4263532554  1.140414756
## 358  -3.3969308178 -1.0261730950  1.433086176
## 359   2.3534875657  0.8018698565 -0.901409818
## 360  -2.8110277470 -1.2315868490  0.912280629
## 361  -4.8121721967  0.2839677359 -1.585037313
## 362  -4.1314385475 -1.4651310876  0.948614455
## 363   0.4023135570  0.1510455689  1.793234547
## 364  -1.7414615045  2.9738334655  2.346251288
## 365  -4.1466821074 -1.8835407276 -1.521245917
## 366  -3.0371213109 -1.0001374046  0.980608526
## 367   0.4006129621  1.3583865532 -0.195912631
## 368   3.2045586811  0.7813068317  0.895887749
## 369  -1.9905175634  0.2373254556  0.021306134
## 370   2.0575462241  1.2054392484 -0.989417479
## 371   4.2652833992 -2.3637097278 -0.519358656
## 372  -0.8497793335 -0.2740759122 -0.986187126
## 373   3.2075393427 -2.4012773759  0.890223085
## 374  -0.3452093894  0.0631030607 -1.471900494
## 375   1.6740644460  0.5012114964 -1.397081369
## 376   3.1414011800 -0.9328825799 -0.987585387
## 377  -1.5810117074  1.6892009731  1.210665567
## 378  -3.2395038556 -0.3879131112  1.793185959
## 379  -3.7575915265 -1.1925324606  1.704189670
## 380  -2.0997335453 -0.1444738960  2.914362133
## 381  -2.8731971144  0.6429082526  0.479178473
## 382   3.2599099791 -0.0652066839  0.955175511
## 383  -1.3673499654  0.8246143350  0.670533533
## 384   0.3726890906 -0.6534604832  1.294377389
## 385  -1.4187905137  4.7109630238 -2.237355884
## 386  -3.4538337331 -0.7260908720 -1.176089301
## 387  -2.3262486132  0.5379411295  0.115382969
## 388  -0.4588861570  0.8340529876 -1.589851516
## 389  -3.4539141799  0.0204235466 -0.788894752
## 390  -0.0880636735  1.9963917469  0.443072562
## 391  -2.3178092968 -0.5443772041  2.841104976
## 392   1.9604778353  0.2828407675  1.769048023
## 393   3.0229976724  2.3346464535 -1.987584575
## 394   0.4617620277 -0.0354262760  2.447255108
## 395   2.1476126858  0.3465558299 -0.328349600
## 396   2.1242315614 -0.5948982234 -3.653908466
## 397  -2.2499492957 -1.1516891319  1.791847224
## 398   3.4184071588 -1.9011849461 -0.521148166
## 399   5.0016192271 -1.3922162544  0.444251802
## 400  -0.0470419540  4.5079300556  0.084083686
## 401  -3.0550200251 -0.4412265955 -1.515880049
## 402  -2.9115198410 -1.7914938980  0.432485923
## 403  -4.9260983451 -1.5303332584 -0.329656642
## 404   3.6179830930 -1.5089342088 -0.904770734
## 405  -4.5286619033  0.5454523002 -1.088131670
## 406   5.3002395384 -2.1270422075 -0.466880860
## 407   1.3733716387 -0.1932096197 -0.838364132
## 408  -3.2836357853 -2.0955076141  1.484345337
## 409   2.6804763819  0.0042547758  0.255157361
## 410   3.3576499013 -0.6189348423 -2.550791997
## 411   0.3769939357  2.2934209968  2.278814045
## 412   2.8811172363  4.8107816177  2.990174491
## 413   3.2280200904  0.3270100122  0.136339440
## 414  -2.9541458326 -1.0968487477 -0.670403296
## 415  -3.0368560705 -0.4080036599 -1.594021463
## 416   2.0404056960  0.4746115129  0.213678195
## 417  -0.3814859133  1.8877546551 -1.044194151
## 418  -2.6307536991  0.7563525854 -1.082772706
## 419  -3.6921795291 -1.2771246294  2.515416946
## 420  -3.6315701293 -1.2579600073  1.051088013
## 421  -0.9397951243  0.0613782483 -1.988747473
## 422  -3.9558994184 -1.7848100461 -0.635009954
## 423  -3.3969308178 -1.0261730950  1.433086176
## 424  -3.5222022565 -2.4562422675  1.520434304
## 425  -4.2059261057 -1.6138722837  0.561780530
## 426  -0.9917260370  0.3261528982  0.376406691
## 427   4.0715109050 -1.8315896180 -0.049038776
## 428  -3.2097261224 -0.5189794482  0.777033399
## 429   1.2131608271  0.1408523123 -1.247276022
## 430   2.5713520633  0.7760260535 -0.025287896
## 431   2.6131926577  0.8320261836  0.874325247
## 432  -2.4819574782 -0.9520979274 -0.600975173
## 433   1.2491469898 -0.6292842130  1.265432778
## 434   2.3167263929  1.3732386393  1.177611796
## 435  -0.6603432502  3.3430898458  1.981027885
## 436   5.0128266295 -0.6852186492  1.098726979
## 437  -0.1642640384  0.5575951809  0.502567910
## 438   0.1889911913 -1.1247184189 -1.168875547
## 439   3.4752605404 -1.5778752397  2.221775016
## 440   0.9937760293  0.6536735067 -0.607852676
## 441   3.2863416643 -1.3258607006 -0.494197450
## 442  -2.2240115954  0.2578061112 -2.350159648
## 443   4.3237403333 -2.0704633107 -0.374767743
## 444  -1.6738666047 -0.8459261083  0.884971084
## 445   2.3705205050 -1.8603833636 -2.721387805
## 446   3.3164873372 -0.4694598860 -2.447130521
## 447  -4.2771610183 -0.5986108869  0.679334426
## 448   0.5972012148  1.1205201262 -0.756888151
## 449   1.5706203407  2.1715333853  0.081546396
## 450   4.1619811890  0.3804379623  1.117516727
## 451  -4.0885796706 -1.1842318066  0.313880163
## 452   1.7343072113  2.8615413442 -0.108279161
## 453  -3.5658230339 -0.0735368050  0.243313149
## 454  -3.7712466623 -2.0970513051  0.292085012
## 455   2.3841149736  1.5478587725  0.322935330
## 456  -3.5525199300 -0.6800779467 -2.207673447
## 457  -0.4220642812  0.3806720757 -1.597090465
## 458   0.5817329569  1.9283245874 -0.146163020
## 459  -0.0252511889  2.7656391552  2.026359871
## 460  -2.6111407012 -0.3763883731 -0.164792326
## 461  -2.8930961347 -0.9467223543 -0.648264126
## 462  -0.1612334442  0.7195699471  1.883178371
## 463  -1.3884888764  0.6700581143 -0.216187714
## 464   2.9894214678 -0.7891907631 -0.149088345
## 465   4.1399880168 -0.4898815459 -0.911436438
## 466   1.0307040411  0.6127204521 -0.447860820
## 467   1.4340769835  0.7509352563  0.659198804
## 468  -4.4386581740 -1.2866467794 -0.622046126
## 469  -0.7712256036 -0.4465087277  1.786249731
## 470   3.5104636936  1.0637926116 -0.426879624
## 471  -0.5281771267  0.8664959745  1.339915088
## 472   3.6673384795 -0.5215473493 -0.093786756
## 473  -0.0739552368  1.5601447007 -0.482508352
## 474   1.2669590936  1.2310699515 -1.891770101
## 475   1.6936389722  0.6467271827 -0.400752859
## 476  -2.9027808166 -1.7618405316 -1.259340761
## 477   2.0100515345  0.3759808768 -1.479512741
## 478   1.8665825190  0.6420421797 -3.237942789
## 479   2.6098545206 -0.7723482800 -1.003478172
## 480  -3.1239322565  0.9819653275  0.595461863
## 481  -3.8605818141 -0.6098458065 -2.007485799
## 482  -0.5399958357 -1.3201537619  3.677801034
## 483   1.8822654934  1.4509464936  0.496416487
## 484  -3.0368560705 -0.4080036599 -1.594021463
## 485   2.9213227759  1.4052432317  0.691703597
## 486   3.2556534996 -1.1092092555  0.606098553
## 487   3.9101037716 -2.1161835832  0.755552547
## 488  -1.4422926608 -1.4645637963  1.337677885
## 489   2.2735919059  1.6363714619  1.502135205
## 490  -1.5273969125  1.0548679335 -0.713118795
## 491   3.1170975407 -0.7469786716  0.906354096
## 492  -0.2536835223  1.2226033497  0.159479422
## 493   4.3237678828 -1.4655175931 -1.467578393
## 494  -0.7392400240  2.5845802771 -1.452828981
## 495   3.8748733383 -1.0786736771 -0.607862274
## 496  -2.2724901794  1.5057321970  0.176512520
## 497  -3.2394430800 -1.3441636717  1.211351744
## 498   2.6287581408 -1.2110904478 -0.222560836
## 499   3.2564696701 -0.8021775734 -2.812765651
## 500  -3.4661657680 -0.0080051507 -0.551385351
## 501   2.7685560760 -1.5498766049 -0.020332157
## 502   0.3202619995 -1.1338053558  1.291793221
## 503   2.4255972863  0.2014955460 -0.225325263
## 504  -1.0575677202 -0.0318694582  2.008806064
## 505  -3.1279749881 -0.9513378096  1.607097848
## 506   2.9327190537 -0.5694202372 -2.057514017
## 507   1.4260965358 -1.7726020561 -0.346083386
## 508   3.1808455194 -0.2380156382  1.622650804
## 509   0.4265529245  2.3369083243 -0.207113923
## 510   1.5177074870 -0.2467970716  2.197548654
## 511   3.0895564365 -1.2909529248  0.159942662
## 512  -0.9583649877  2.9081720564 -1.105305277
## 513  -0.2491658313  0.3196833740 -0.330829009
## 514  -3.0068067034 -0.8879745381  0.730456777
## 515  -0.4822743000  1.7436935307 -0.654040467
## 516  -2.8261133022 -0.7909245171 -1.491228077
## 517   3.1320410497 -1.3231254145 -0.254185263
## 518   0.0707081809 -1.0740097289 -0.025631743
## 519  -2.4522772620  0.0188788587  0.810644393
## 520  -2.9633519172 -0.6192623403  2.236373928
## 521  -0.6367438154  0.2401854839  1.319318793
## 522  -0.2456807790  0.7866282095  0.197430619
## 523   2.3850390674 -0.3928969356 -2.663557218
## 524   2.1154450663 -0.7001685773  0.742394427
## 525  -2.9070672779 -1.9132938171  1.252099550
## 526   3.5764256688 -2.2282373455 -0.747570579
## 527  -1.1823428605 -0.9879733395  0.836786036
## 528  -2.8824241041  0.7098248854  0.084608262
## 529  -4.7242082505 -0.2850161035 -1.572757562
## 530  -2.5052586493 -1.1898908699 -0.072039736
## 531  -2.2416684444 -1.2295708822  0.790921877
## 532  -0.0683541708  0.7863299855 -1.802092027
## 533  -1.1345865836 -0.8756485144  1.168513689
## 534   4.2799557134 -1.2629435176 -1.418544904
## 535  -0.7359387871  2.5069738272  1.151486968
## 536   2.4292012873  0.2809132111  0.340911455
## 537   4.1305709390 -1.5935317210  0.679829079
## 538  -2.1818639501 -1.6790105917  1.567356062
## 539   3.9362388918 -0.1011717480 -0.877971360
## 540  -1.9941640369 -1.2412989374  0.615488029
## 541   3.9071010590 -2.0011962881  0.709868666
## 542   2.2351600934  0.7713631017 -0.637477646
## 543   2.9785768003 -0.0746263486  1.465582742
## 544   0.4810588469  0.1751264774 -1.099030328
## 545  -0.6756192475  0.3084526136 -2.514734952
## 546   1.7316954370  1.5105739118  1.358171758
## 547  -4.6905436787 -0.2577295864 -1.608471463
## 548  -4.0555768758 -1.0965798862 -0.261015769
## 549  -4.6467648285  0.0624316902 -0.064868524
## 550  -0.0535973950  1.4430179566  1.716292069
## 551  -4.2809111732  0.3038538374 -2.335900585
## 552  -0.9417098046 -0.2432832255  2.129088333
## 553  -0.4511480383  0.3569258791 -1.495725317
## 554  -3.1564650742 -0.3513862604 -0.196424923
## 555   1.3286074025  1.4799110637 -2.106005110
## 556  -1.9593326488  0.8225912011 -0.567720753
## 557  -3.3417283633  0.7646941178  1.109643629
## 558  -3.0512347604  0.8353054214 -1.971645823
## 559  -1.0129010291  2.1213382170 -0.448265034
## 560  -2.4242170535  2.4324604494 -1.229754065
## 561  -0.0400413525  2.6691918008  1.031036764
## 562  -3.1427301890 -0.4311498900  0.765366809
## 563  -1.1426702392 -0.6444076591  0.011559052
## 564  -3.6535920726 -0.8099582269 -0.712758487
## 565  -2.7478608636 -0.2654167844  0.167591566
## 566   4.0461486716 -0.0741032802 -0.757628911
## 567   2.1480362863 -0.4186265454 -0.680948845
## 568   0.0545826855  1.0654844500 -0.876342140
## 569  -2.2051924585 -1.7301975962  0.297526876
## 570  -2.2162099680  0.1619091676  0.132378571
## 571   5.0990699556 -1.5242942554 -0.360309675
## 572   4.2183551618 -1.6341913590  0.052282556
## 573  -2.8321601247 -1.9793908803 -0.077245269
## 574   1.0636239844 -1.7383923268 -1.679475595
## 575  -2.4769943592 -0.0607765875  0.630248911
## 576  -2.4923855490 -0.4498488543  1.323912584
## 577  -4.2203484592  0.5393532504 -1.044526640
## 578  -2.3795790356 -1.0274787323  1.915741535
## 579  -2.4453453885  0.3689584757  1.505368716
## 580  -3.9899684211 -0.4227349343 -0.642414608
## 581   2.1506420791  1.2095372286  0.834981783
## 582   3.3056555105 -1.4844933612 -0.228139644
## 583   0.6942289963  2.5813922142 -1.353641115
## 584  -1.9610957985  1.3777781322 -0.322872775
## 585  -1.5104363012  1.5884727831  1.787726523
## 586  -3.1067010992 -1.0000362343 -0.903058769
## 587  -4.1144773028  0.0607348017 -0.319357643
## 588  -3.0077984749 -0.7346887812  2.001884233
## 589   2.5342734742 -1.0879403104 -1.105822266
## 590  -0.0549539481 -0.5918934999  0.597865475
## 591   0.0132077426 -0.6734168007 -1.444538685
## 592  -3.5821796195 -1.0343162594  2.350574001
## 593  -4.3445165243  0.8168717587 -0.349659185
## 594   3.0487461770  0.6480428468  0.433472985
## 595  -0.0559817308  1.6535746291  1.210190510
## 596  -4.4747146219  1.1409011039 -0.987677723
## 597  -2.6275904125 -1.3062288503  0.276145679
## 598   0.2617046409  0.7675586607  0.062805219
## 599  -0.6972407652  1.6663100464 -1.729623300
## 600  -1.6556517725 -1.5128312348  1.267095171
## 601   2.7118462803 -1.6119634019 -1.514842541
## 602  -3.4258308846 -1.4323253429  0.908408781
## 603   3.5066626323 -1.0414142230  0.401464558
## 604  -3.4758991755 -0.6811070991  0.681297396
## 605   3.5003286628 -0.5407909611  0.927330495
## 606   3.7753295384  2.8546121032  1.187678083
## 607  -2.2836918857  1.2566999040  0.787192279
## 608  -1.6865330498  0.6946586579  1.606298189
## 609   2.7834426184  1.3260677193 -0.378532140
## 610   2.4002455721 -1.3434955683 -2.047597741
## 611   2.9785768003 -0.0746263486  1.465582742
## 612  -1.5174464540 -1.8975044717  0.436200537
## 613   4.0073101873 -1.1768207605  0.226859216
## 614  -5.0452351918 -0.1184454688 -1.609730481
## 615   4.6241195010  0.0926966440 -0.099416046
## 616   1.0553680322 -0.6861065105  0.318482706
## 617   3.1316590398  2.5288487796  0.574970299
## 618   2.8457900424 -0.2476128564  0.218986425
## 619   3.2045586811  0.7813068317  0.895887749
## 620   2.8407624261 -0.5438373509 -3.023589410
## 621   2.1056340258  2.2799785546 -0.223010429
## 622   0.1494797373  0.4028279555 -0.071653712
## 623   3.3915227840 -0.2676111368 -1.702068216
## 624  -1.1170206984  2.3150848552 -0.513300510
## 625  -0.3492362306 -0.4999065111 -0.762283612
## 626   2.7028935190  0.4418667119 -0.940768386
## 627   1.4086210303  0.6150282853  0.285845685
## 628   4.3978033382 -1.8611449355 -1.987029002
## 629  -3.3495154427 -0.1399922722  1.558126770
## 630   3.0963401858  1.7914210562  1.462965973
## 631  -2.0870247598 -0.8437038107 -3.081624845
## 632  -0.3573540739  1.4915425048  0.401151732
## 633  -1.1669444703 -0.1569736841 -1.050494261
## 634  -0.1822427364 -0.7003299903  2.169782103
## 635   3.1000045879  1.2766144138 -1.707122879
## 636  -1.4641750876  2.1497551993 -0.988587317
## 637   0.2475165722  1.8924101783  0.232765735
## 638  -4.1314385475 -1.4651310876  0.948614455
## 639   3.2726188317 -0.4279702041  1.149627693
## 640  -2.5373736587 -1.5571126803  2.625327328
## 641  -3.4755402772 -1.3143799590 -0.535819507
## 642  -2.7233563842 -0.8765589489  1.384920072
## 643  -4.2119142026 -0.3876198308 -1.514263271
## 644  -2.8529254950 -1.6400346458  2.252865189
## 645  -4.1337422792  0.0225733282 -1.508667727
## 646  -0.6083383719 -0.1295572705  0.743311433
## 647   2.9909083090  0.8604728929 -0.710221375
## 648   0.9751789497  2.0859833984  0.067109035
## 649   2.7119370961  0.2261764268  1.748618638
## 650   2.5145650785 -0.4957387603  0.998535638
## 651   0.4742783623  1.2808602107 -0.279661785
## 652  -5.0695361220 -1.0599285588 -0.327722070
## 653   2.5062151993  2.0410115409 -0.891268572
## 654   2.4946841297 -2.3963276885  0.048986939
## 655  -1.3831100431  0.4108809613  1.699607187
## 656   2.3234377427 -1.4556284055 -0.997721434
## 657   2.3634324361  0.6864308006  0.384292139
## 658   4.7251483693  0.6563420727  1.897149223
## 659   0.8276446403 -0.9505123327  2.658173415
## 660  -0.4324660132  0.1314697669  0.031537503
## 661   2.9304165613 -1.7110739886 -0.136572398
## 662   2.1517410637  0.2175552847 -0.296244099
## 663   4.7119209699 -0.6290261485 -0.363138329
## 664   3.0200889983 -1.4755108603 -2.224053685
## 665   2.7172644592 -1.0970578347 -0.973571592
## 666  -0.3594590949  2.5603476376 -1.448080919
## 667   1.9353399306 -1.1602739770  0.283654303
## 668   0.1819619266 -0.2814042208 -0.208132851
## 669  -3.4597118884 -1.6313228189  0.665017024
## 670  -2.1598952179  1.3563048929 -1.332006714
## 671  -1.5393458905 -0.1704995171 -1.500382168
## 672  -3.6945965695 -2.5551080114  0.333102126
## 673  -3.1429458440  1.2603295200 -1.569005695
## 674   1.8685080772 -0.0965505928  0.820147948
## 675   3.7658364677 -0.5337913166 -0.828773926
## 676   2.0395984044  0.5194493576  0.551061954
## 677   1.7536117917 -0.0257760695  0.143737096
## 678   1.6873557966  0.9996824802 -0.025941677
## 679   3.2509114403 -2.0549024589 -1.864659354
## 680  -0.9911057861  0.2369561490  0.107391889
## 681   3.6076561599 -0.1095651065  1.112323884
## 682  -2.4701280973 -1.4374322472 -2.011700853
## 683   0.5313171844  1.4688682942  0.187960813
## 684  -0.0708525551  2.0067041559 -2.193886697
## 685  -2.5949342469 -0.5345156182 -0.630888057
## 686  -4.2752355402 -1.4129974302  1.122490317
## 687   1.8947302734  1.8391554996  0.304976783
## 688  -1.6061412501 -1.0431182647  1.339941221
## 689   1.7670459936  0.1007257538 -0.137656898
## 690  -0.6161718996 -0.1231209252  1.034182806
## 691   3.2678999651 -1.0353363335  0.157345220
## 692   3.8008383531 -1.0851132599  0.824594986
## 693  -2.9117355613 -0.8125124565  0.941510385
## 694   2.7988796050  2.3224462687  1.362785872
## 695  -4.0307410800 -0.7353238424 -1.123450375
## 696   1.1351300145  0.8262269184 -0.873855689
## 697   1.8223625493  1.0375098007  0.320116479
## 698  -4.5240647883 -0.9128717217 -1.172180496
## 699   4.0676322326 -1.3037283678  1.044627680
## 700  -2.6729377616  1.1541977835  2.129077032
## 701  -3.3167412283 -0.7206107820 -1.244958762
## 702   4.1468940597 -0.7979338454 -0.470467141
## 703   4.0328665033 -2.2055839746 -0.469780793
## 704  -0.4899983176  3.4378210417 -0.169248591
## 705   3.6442872415 -0.5399919613  0.896433782
## 706  -4.7674304660 -0.0927498149 -2.701533787
## 707  -0.2878583572  4.5649931740 -0.327398185
## 708   1.3023736277  1.7083868510  0.848313769
## 709  -1.1951510769  3.4369610224 -1.093253037
## 710   2.2850663277  0.5818255633 -0.140861567
## 711   3.3124689890 -1.1719849645 -3.116378521
## 712   3.2708461125 -0.1387393192  1.395603756
## 713   1.1343746765 -1.0104163801 -1.606139986
## 714  -4.6334677606 -1.9465675154  1.863312575
## 715   3.0375887997  0.4007863542  1.354639326
## 716   2.4261769688  1.6582697537 -1.699176027
## 717  -2.4178700583  1.0504027068 -0.434811792
## 718  -2.8727123357  1.1961525412 -2.537793716
## 719  -1.4887908262 -0.9140858399  1.875510128
## 720   3.4069769699 -1.5430689816 -0.642562755
## 721   4.0759771216 -0.9141829089 -0.622136294
## 722  -3.0472751614 -1.1093159777  1.562186271
## 723   3.5307061441 -2.6792229623  0.235975355
## 724  -3.4967738520 -1.0671361232  0.230759769
## 725   3.7333908257 -0.3434423227 -0.224120338
## 726   4.0499815590 -0.8490810300 -0.885250852
## 727   0.3815681165  0.2662545587 -1.905673833
## 728   3.2726188317 -0.4279702041  1.149627693
## 729   4.2585690868  0.0686396569  0.149442587
## 730   3.7412131896 -1.4384707415  0.387075035
## 731   3.4030129160 -1.0484265936  1.616191012
## 732   1.8599448696  2.1902409126 -0.292763760
## 733   0.8546716575  1.8428402832  0.397965242
## 734  -2.0902063678 -0.4862993601 -0.460826109
## 735  -2.6756961419 -0.0628826459 -2.486974799
## 736  -2.3306438506 -0.9195002855  1.644844707
## 737   1.2956374804  0.5782397003 -0.340909847
## 738   3.7133423333 -1.2340226123  0.413040196
## 739  -2.4200640501 -0.2061203889 -0.889594981
## 740   3.2613221146 -2.0174112934 -0.914328302
## 741  -1.0028525553  0.7641398448 -0.174155894
## 742   3.1426258613 -1.0288524138  0.594466777
## 743   4.2115862177  0.1053838359  2.258731860
## 744  -0.4849797017 -1.3105253954  1.190971364
## 745   0.2490536025  1.7675931041 -0.699521853
## 746   2.6660002759 -0.5903183994  0.606444497
## 747   3.3801080429  0.2988296856 -1.402336304
## 748   0.8678797763  3.1609891397  1.309659500
## 749  -2.7043651382  0.5630875958  0.400178429
## 750  -4.1052366368  2.0087792029  0.864439192
## 751  -4.7483865393 -0.0492708482 -1.195081026
## 752  -2.8140641369  0.4286119429 -1.747251019
## 753   1.8314324677 -1.5682267008 -3.532622439
## 754  -3.0517068191  0.0022166365 -1.753498535
## 755   3.2358192468 -0.3526061610  0.194999292
## 756   2.5858209756 -1.9026046175 -2.099090619
## 757  -3.5623266409 -1.2013794443 -0.397784600
## 758  -3.8271377451 -1.6126324226  0.076403899
## 759  -3.8377455287 -1.7202060100 -0.007446082
## 760  -3.2744647345 -1.0443370850  0.570549078
## 761  -2.8708198523 -0.9361309214  2.758636956
## 762  -3.4968086549  1.3018029887 -0.777502632
## 763   2.1805051439  1.8785866477  0.209578880
## 764   0.8321085969  1.7607172368  0.256848477
## 765  -0.1460326111  1.6723295139  1.973991855
## 766   3.3965949468 -1.1282661634 -0.702224025
## 767   3.1676969108 -0.3612811984 -1.668752362
## 768  -2.4723814271 -1.2480139472  0.346550565
## 769  -0.8816492246  0.6982671473 -0.011643472
## 770  -0.3231637903 -1.1064175078  0.472535154
## 771  -0.0320523289 -0.2595086237 -0.644581343
## 772   2.1148777618  1.4023899760 -1.341774660
## 773   2.5321567584  1.9016471482  2.228301674
## 774   3.9305142946 -1.1828196403  0.411668481
## 775  -2.9015497559  0.2957778137  0.709808204
## 776   1.3777086450  0.5755246029 -0.301712054
## 777  -3.0862365582 -0.3349867159 -0.705715862
## 778  -1.0214657499  0.4263356343  0.692289970
## 779   4.2888592820 -0.3717992156 -2.086125679
## 780   3.4933739492  0.0036869728  0.237156233
## 781  -1.3925978009  3.1242864023 -0.475363211
## 782  -2.3356007678 -1.8239300819 -2.090506726
## 783   0.9471594946  0.2174457817 -0.021215427
## 784   1.3767751685  2.3324152076  1.464665753
## 785   3.6871752768 -0.7372931601  0.522173789
## 786  -2.3409605830 -2.0680100796  1.344186465
## 787  -3.0309739745 -0.4106862910  0.588467510
## 788   2.5494796113 -2.1559648634 -0.073106759
## 789   2.1671475018  1.6920871752 -0.279692748
## 790   4.2829635199 -1.6711888647 -0.728744177
## 791   0.6904390407 -0.2061143715 -2.352701111
## 792  -4.2010488091 -1.1575723326 -0.287439090
## 793   0.7487707012  1.4393459189  2.086121725
## 794   1.0088800202  0.3994799649 -0.688665685
## 795  -2.7898676111 -0.9520030757  1.919835049
## 796   3.5914579683 -0.9101264017 -1.282695937
## 797  -2.7387974002 -0.7423194968  1.062082404
## 798   0.9270360403 -0.3504657251 -1.410268251
## 799  -1.1345865836 -0.8756485144  1.168513689
## 800   3.9362388918 -0.1011717480 -0.877971360
## 801   3.3700574486  0.6755053541  1.266056811
## 802   1.4636729771  1.8270624226 -1.506864475
## 803  -4.0240566141  0.7202528444 -0.277286367
## 804   3.5088497183  0.4076987617 -1.176012146
## 805  -4.4181821244 -1.9126405356 -0.093116019
## 806  -2.3341418702 -0.3725189774 -0.947176262
## 807   1.3745342913 -0.2525829966  1.925216497
## 808   0.1120011575  1.8895412881 -1.552048967
## 809  -3.2211834764 -1.0888631222  1.772109083
## 810  -3.8628019068  0.1842685721 -0.102629758
## 811   3.6531190326  0.8925595702  0.654846605
## 812  -0.2564873002  0.9070370281 -0.007598237
## 813   4.3561834070 -0.4154366069  1.214708723
## 814  -3.3597736182 -1.1378468808  0.462033880
## 815   1.4876162149 -0.0243929194 -0.074777687
## 816  -2.0530665573  0.2715250595  0.005046854
## 817   0.2471419135 -0.0194372710  1.734853315
## 818  -1.9093338872 -0.9568516760  1.237849585
## 819  -2.6919527621  0.4379868405  2.549897741
## 820   3.3909340682 -1.8897591959  0.245145111
## 821   4.6167147437 -1.6764509130 -0.727985468
## 822   2.6511678753 -0.9549311470  2.045211988
## 823   1.2437149587  1.9268530875 -0.434333794
## 824  -0.7855151833 -0.2251446048 -0.032553845
## 825  -3.1079866793  0.3453754984  1.090348765
## 826  -0.6599481472  0.3791680884  0.309308558
## 827   0.7506349783  0.7904782225 -2.606283209
## 828   4.9613839422 -0.6231907815  0.943111431
## 829  -3.9593527200  0.0565225600 -2.368272530
## 830  -0.9795250129  1.6863161901 -2.312290363
## 831   3.0263877277  2.1709289726  2.275580078
## 832  -4.1135793779 -0.8480200740 -1.883024878
## 833   1.7940896426  0.6493759230  1.429953049
## 834  -3.9549411663  0.7060055425  0.124679493
## 835   3.7000313637 -1.6021788738  0.512218341
## 836  -3.6764165149 -0.8290772167 -0.312068608
## 837  -1.6526988676  2.6164578685 -0.761987349
## 838  -4.3715025062 -2.4111661105  1.311252611
## 839   0.3002707148  0.6854878096  3.793967894
## 840   0.4845579675 -0.2522670676  2.177113426
## 841   0.4028095207  0.5720174415 -1.156846733
## 842   0.1384679196  0.5174178630 -0.441990530
## 843  -5.3758140903 -0.5997364014 -0.418008845
## 844   1.4437587579 -0.7374139296  1.203104711
## 845  -2.3350519699 -0.6459896876  2.021821555
## 846   1.2420370293 -0.5046826115 -2.704082314
## 847   2.4420762989 -0.7863372936 -0.435369640
## 848  -3.2331890105 -0.6250562172  0.435712436
## 849  -1.1095233011 -0.1116485763  1.482367889
## 850   2.8762423627 -0.3813394823 -0.024711396
## 851   4.2371060019 -1.3634391389 -0.689032022
## 852   0.4909149224  2.6020052614  0.501552159
## 853  -1.4696648534 -0.2692670117  1.346170352
## 854   4.6060578732 -1.3013754786 -1.640220065
## 855  -1.8732391888 -0.2205722882  2.678391260
## 856  -2.1079104267 -0.6014551193  0.680731041
## 857  -2.7459308086  0.2210314003  1.168271752
## 858  -4.4526480915 -0.9365841533 -0.463401722
## 859  -2.9782552261 -0.8607117929  0.241537321
## 860   4.7119209699 -0.6290261485 -0.363138329
## 861   4.2566581683 -1.0357849906 -0.796860663
## 862  -1.6971828172  3.6485685952 -0.733456972
## 863  -3.8458557547  0.6761691547 -0.808374811
## 864   0.9306342326  3.2216589215  1.587953530
## 865   2.9088992217 -0.8747859520  0.170181017
## 866  -2.5198677490  0.0237581850  0.730395877
## 867   0.0614086047  2.3403950050  0.690533282
## 868   3.8743888645 -2.2356028409 -0.029414841
## 869  -4.7832147363 -2.5008062204  1.134888882
## 870   3.0366948807  0.5024741609 -1.315797012
## 871   1.7306595214  1.7902528686 -0.263622390
## 872   2.8700429935 -0.2533317150  0.232510250
## 873   4.0614987850 -0.0221085319  0.120685405
## 874  -0.2636129051  0.6857675630 -0.804636857
## 875  -2.2434717264 -0.0497279434  2.140463708
## 876   3.2198154113 -1.2966103580 -0.034111480
## 877  -1.8427134482  0.3722366841 -0.409689735
## 878   2.6703790426 -1.5474988168  0.281829887
## 879   1.0641478927 -1.3901242280  0.356628814
## 880  -0.9141208729  2.0487746548 -0.059612685
## 881   3.8269545287 -2.2448482532 -0.075724402
## 882   3.5528284631 -1.0947341046 -0.878078575
## 883  -3.5387263831 -1.1291308922  0.202241509
## 884  -2.7338341907 -1.2087002424 -0.857185727
## 885  -0.5167769539 -1.6997623294 -1.151502953
## 886   3.8017925800 -1.2630032592 -1.547737318
## 887   3.0397899605 -2.0506333726  0.428772375
## 888   0.9275699273  3.9151168740 -1.275568577
## 889  -1.0646266932  0.1004742160  0.004335313
## 890   3.9046241384 -1.0281617296  1.048933879
## 891   2.8750201549 -1.6210774280 -0.023145731
## 892   2.2264773626 -0.9690843719 -1.079910385
## 893   2.3850390674 -0.3928969356 -2.663557218
## 894   0.8042400322 -0.7495261975  0.764870338
## 895   3.5798669851 -0.1342693473 -2.150080423
## 896  -3.6579387406  0.3237943345 -3.033220721
## 897   3.4279339021 -2.3391251757  0.912865496
## 898  -1.5788047192 -0.7426606655 -0.030717574
## 899  -2.5105679089  0.3678683218 -2.277170313
## 900   4.1129225326 -0.4029674525  0.626871574
## 901   0.5497025021  0.7272796714  2.174421905
## 902   3.3325130940 -1.8592020017 -0.148864166
## 903   4.2211653464 -1.8028422031  1.421504707
## 904   3.2075393427 -2.4012773759  0.890223085
## 905   2.9775652892 -0.0613375296 -0.977952671
## 906   2.3874134196 -0.8951361492 -2.227403847
## 907  -1.0899115541 -2.2414459185  0.418787922
## 908   1.2747133662  3.0085815352 -1.471565889
## 909  -3.6066035453 -1.4372298152  2.775223407
## 910   3.2613221146 -2.0174112934 -0.914328302
## 911  -3.4481907294  0.0368559083 -1.578327897
## 912   2.6994056639 -0.7667567443  0.759489500
## 913   2.7646590680  2.4674506804 -0.880584345
## 914   2.3502159104  0.7016436540  0.788246800
## 915   1.6620440717  0.8457215902 -0.676203775
## 916   2.2688396248  2.4056297390  1.894149391
## 917   1.3878163319  1.1380717533 -0.940202210
## 918   3.3314063384 -2.1327100990  0.592210031
## 919   4.9862283727 -1.2286810126 -1.023792813
## 920  -2.2513200859 -1.1425090427  3.099877194
## 921   0.3394307214  1.0881729612 -0.923172748
## 922   2.1376081881 -1.0101477558  1.248774139
## 923   0.5642029489 -0.3874576977  0.667001653
## 924  -2.4796948611 -1.3979283590  0.474631799
## 925   1.5573242276  0.3342499671  2.014083426
## 926  -3.0647612131 -1.1981129670 -1.368437085
## 927  -2.7504531671  0.5481714104 -2.153297170
## 928  -1.2586804266  0.9999639598 -0.178991765
## 929  -1.7165868633  0.3544747524 -0.181194418
## 930  -3.3800106489 -0.4525605035 -1.246015068
## 931  -3.4214171478  1.6557675756 -0.227517882
## 932   1.2451780114  0.6113311685  1.423101415
## 933   2.0002536924  1.2574676918  1.015633890
## 934  -2.3445236482 -0.3634377184 -0.971782439
## 935  -0.5067598139  1.7759838008 -1.327203204
## 936   3.6985662352 -0.1466018316  0.857211606
## 937  -0.0465084578  1.2606163410 -2.200242310
## 938  -1.9647009764 -0.2326307803  0.455284239
## 939  -1.7893881789  0.8840165653 -0.748582168
## 940  -0.6767733142  1.0095320165  0.521802653
## 941   2.1841246774  0.4233916019  0.705326467
## 942   4.4060553171 -0.9508173916 -0.630023477
## 943  -2.2531102316  0.6022253449  1.089989109
## 944  -4.2026976458 -0.1867143594 -1.243626604
## 945   1.5454266145  2.2905046818  3.019646386
## 946   3.5125749918 -2.5219361424  0.114809581
## 947  -3.1171037848  2.0604097601 -1.197929692
## 948  -2.2565662648 -0.0391763172  2.947458425
## 949   1.4496827994  0.1869262091  1.235234359
## 950   4.1212805021 -0.8628994499  0.017047256
## 951   3.3713096692 -2.2361868363 -1.696141608
## 952   3.8647754176 -0.8588172007 -0.830785981
## 953  -0.4899983176  3.4378210417 -0.169248591
## 954  -1.9632525862 -0.0418444170 -0.582831200
## 955   1.8838335832  2.6835256897  0.211735215
## 956  -3.8389286331 -0.9252485990 -0.391942759
## 957  -6.0118737628 -1.5488933859  0.282135149
## 958  -3.1409268052 -0.4377883559 -0.188169605
## 959   3.2002412419  0.9680757429  1.297290249
## 960   4.7309477928 -1.0541650670 -0.417828219
## 961   0.3486107686  0.4057482364 -1.377003696
## 962   3.7601231387 -0.9953373633 -0.914305327
## 963   4.9887592718 -0.0995617888  0.118206808
## 964   4.3323946280 -1.7266396879  0.968880746
## 965  -3.5052758304  0.3215733923 -2.627098755
## 966   4.1266570656 -1.7485814239 -2.118740239
## 967  -4.2135674369 -1.0336257675  1.292199439
## 968   0.3101811905  2.8147209241  1.204007361
## 969   2.5703242146 -0.4416255937  1.092796667
## 970  -1.9375785828 -0.3072451418 -1.775608174
## 971  -3.6945965695 -2.5551080114  0.333102126
## 972   2.2384191168 -0.3517717952 -0.486699138
## 973  -2.0339434607 -0.2681048543  0.038629901
## 974   2.7229853458  0.5511816407  0.990397302
## 975  -0.1256330016  0.6867496730 -0.143194778
## 976   1.8578304992  1.3104274209 -1.414292273
## 977   3.3453441853 -1.4655546689 -1.557602638
## 978   1.4270994701  1.9068515910  1.697732428
## 979  -2.8859078068 -0.9375422560  1.739108665
## 980  -1.2963685880  2.4645296990 -1.046765392
## 981   1.7612061209  0.4784207601  0.590280418
## 982   2.8813572169  0.8236879533 -0.096667584
## 983   4.0759771216 -0.9141829089 -0.622136294
## 984  -3.0263204839  0.7210300219  0.223694730
## 985  -4.4687533596  0.0862408909 -1.790493867
## 986   3.6401353035  0.8591862347  0.962114843
## 987  -2.1471619925 -1.3407132458 -2.775426946
## 988  -1.0341727860  2.0570201918  1.716142319
## 989  -2.5837668500  0.5911354563  0.006527726
## 990  -3.6786472071  0.2630069039 -1.723539346
## 991   1.5329476147 -1.4533313368 -1.288046208
## 992   3.8452956354 -1.7690502509  0.500538082
## 993   0.0486530395 -1.6075189093 -0.331948349
## 994  -3.2015116384 -0.9718229330  0.576233796
## 995  -1.2038744781 -1.7197464092  1.554740510
## 996  -2.7435649434 -0.3412781286 -0.372044734
## 997  -3.6956810362  0.0449537237 -1.382487140
## 998   1.2988876444  4.7444432425 -0.868551206
## 999  -3.3967370523 -0.7542901640 -1.860836262
## 1000 -3.0327829915 -0.3161307901  1.936781678
## 1001  2.4184052925  0.3630154417 -0.173358540
## 1002 -1.8855878092 -0.9100141830  0.661688058
## 1003 -2.6811497641 -1.1370737788  2.896846484
## 1004 -4.0261284456 -0.5557595580 -2.312457182
## 1005  3.2189388029  1.9307424947  1.854181221
## 1006  4.1524596955 -1.1811203092  0.249543567
## 1007  4.3309046935 -1.2115262799  0.318189325
## 1008  2.9284060818 -1.9465699955 -0.257782239
## 1009 -2.8211997901 -1.3059194524  1.835847338
## 1010  3.9888325987 -1.8923549644  0.054106400
## 1011  0.2081266385 -0.3711588482  2.681495770
## 1012 -2.5705402192  0.8668285160 -1.966096896
## 1013 -3.1199580716  1.0756217470 -0.938084275
## 1014 -2.2256275748 -0.9783597403 -1.937600404
## 1015  1.0486669646  1.4761182824  0.156649774
## 1016  1.6813017932 -0.3123975228 -0.604036577
## 1017 -3.4786184759  1.5325691839  0.081402029
## 1018 -3.1901969700  4.1331735326  4.796162985
## 1019  3.0106222261 -1.6952817146 -0.406909355
## 1020 -3.5678063639 -1.0737466590 -0.107957283
## 1021 -2.5183875675  0.9455338358 -0.398277691
## 1022 -1.2640198126 -0.8504772764 -0.058559720
## 1023  3.6891210358  0.0065011544 -1.467647958
## 1024  0.6451106512  1.6957269519  3.046693985
## 1025  1.2011742793  1.7216433398  0.487199940
## 1026  3.8452956354 -1.7690502509  0.500538082
## 1027 -5.0074264124 -0.1272171741 -2.324474397
## 1028  3.6933531945  0.8244061940 -1.545428252
## 1029 -2.8211502433 -0.7179018599  1.116033626
## 1030 -3.6845294216 -0.6719266897 -0.061004661
## 1031  2.3677841467 -1.0396860943  0.434178212
## 1032 -1.1399195656  1.6989767830  1.257607367
## 1033 -1.3539439662  0.8584716409 -2.169491664
## 1034  3.3390556714 -0.1548749629  0.096864267
## 1035  2.9999422061 -2.3687060751  0.790820259
## 1036 -0.1671950460  5.3579740881 -0.785932002
## 1037  2.1835240055  1.6713011834 -0.588017309
## 1038  2.2249822436 -0.7394259970  0.487116725
## 1039 -3.0763759055 -0.3210220547 -1.657175133
## 1040  0.6942486822 -0.4573685698 -2.814370058
## 1041 -3.4693896937  0.6533384281  1.036380062
## 1042  4.8648894764 -2.5406410403 -0.128152952
## 1043  3.6967984616  2.5309418995  2.544872439
## 1044 -3.9110885952 -0.9395109782  0.676727795
## 1045  2.6778059726 -1.7636647722 -0.769817138
## 1046 -3.4481907294  0.0368559083 -1.578327897
## 1047  3.5995213802  1.2673196976  1.573284020
## 1048  1.9857008561  1.8011654097  0.942247246
## 1049 -3.1810545120 -0.4918419677 -1.295228217
## 1050  0.4589786133 -0.0419237820  0.230736481
## 1051  4.8873626852 -1.4439406367 -0.182274908
## 1052  4.0571639812 -0.8449507248 -1.283208896
## 1053 -4.2977532620 -0.4600611910  0.706460152
## 1054  0.4563280983 -0.2331041246 -0.734372478
## 1055 -3.8996336985 -0.7082382282 -0.920783606
## 1056  2.8026857702 -2.0122040804  0.185847750
## 1057 -0.0083645416  2.2086374086  1.859460125
## 1058  3.6246430930 -0.6364422011  0.413647254
## 1059 -4.1926593356 -1.8027166775 -0.356357611
## 1060  4.2954323915 -0.9556412432 -0.448091500
## 1061  1.3860234794  2.1589569465  1.003172559
## 1062 -4.1359166347 -0.7811816049 -0.219106680
## 1063  3.8067793434 -0.4682803147 -1.534405762
## 1064 -3.8261231094 -1.6801448638 -0.370922789
## 1065  3.4996967937 -1.5923849333 -1.786674268
## 1066  3.7805521213 -1.0103923401 -1.895474294
## 1067 -1.3239575489 -0.7948276784 -1.003646094
## 1068  2.6287581408 -1.2110904478 -0.222560836
## 1069 -2.8170408696  1.0081494116 -0.794204057
## 1070 -2.3426286849  0.3709963868 -2.807738075
## 1071  1.3222307112 -1.0779911155  0.822742099
## 1072  1.7555770258 -1.0767965696 -1.423092551
## 1073  2.8030424235 -2.1053800596  1.854392559
## 1074 -4.8026898868 -0.9158927498 -0.525570048
## 1075  2.5423430444 -0.1861242985  0.772488993
## 1076  4.7265228496 -2.3933075883 -0.199418388
## 1077  0.7128388955 -0.8302830003  1.736176048
## 1078  0.4375636259  1.4501014258  2.304052088
## 1079 -0.6734864495 -0.0206379143 -0.088911419
## 1080 -2.9139586365 -0.9702510410  0.176054604
## 1081  2.9782415057 -0.9868828448 -2.266036155
## 1082 -3.1550278181 -0.2412869085 -3.253413385
## 1083 -2.1818639501 -1.6790105917  1.567356062
## 1084  1.4267922361 -0.6070160549  0.281949706
## 1085 -0.0144130929  1.2586143477 -0.316596216
## 1086  3.0258974914 -1.5625331297 -0.570541351
## 1087  2.8643466290 -1.1215768913  1.303900883
## 1088 -4.4719763416 -0.7186833247 -2.642979180
## 1089  4.3978033382 -1.8611449355 -1.987029002
## 1090 -3.6271679216  0.1521041579 -0.278425790
## 1091  1.5459033249  0.7420248919  0.057343207
## 1092 -2.7460890911 -0.3624283834 -0.674052646
## 1093  1.1399472344 -1.3735800199  0.966185125
## 1094 -2.1567344862 -1.6242278397  2.024837983
## 1095  0.4684974832  2.0749839897  2.138049911
## 1096  2.4898716612 -0.3235904867  0.043445759
## 1097 -2.4225949186  0.6833822096 -0.454995907
## 1098 -0.6029672413  1.9072797750  0.247605078
## 1099 -0.7833876192 -1.8950696227  0.666970292
## 1100  0.2312721140  0.9446316908  0.949826156
## 1101 -0.1652118101  3.1163750452  1.451609005
## 1102  2.5500826133  1.9637905202  1.591123303
## 1103 -2.8221071504 -0.2739578097 -0.850260594
## 1104 -3.5725967378 -2.8130528030  1.561443168
## 1105  3.0190633583 -0.2565806884  0.023919326
## 1106  0.6943563252  1.7035256657 -1.111690113
## 1107 -4.6140910551 -0.4952059806  0.056093313
## 1108 -2.4851676011 -1.7835997325 -0.045150825
## 1109  4.2092021349  0.1183356982  0.978074921
## 1110  3.4701225137 -0.6299664269 -0.178197909
## 1111 -0.1462300814 -0.4145580729  0.657712954
## 1112 -2.7572511512  1.6835260889 -0.075091822
## 1113  3.2079194869 -0.8621545114 -2.212954837
## 1114 -3.2703833477 -1.8318789285 -0.040975723
## 1115 -0.6039434506  0.7974860658 -0.030849135
## 1116  2.1451019118  1.0065526156 -0.077088803
## 1117 -2.4004039470 -1.3874159816  0.558579759
## 1118 -3.1041550918 -1.7925424772  1.392975717
## 1119 -2.9439788308  0.7092553623 -1.390159479
## 1120  3.8419901473 -0.4715901782 -0.035140216
## 1121 -2.9027276939 -0.2984122059 -1.046130369
## 1122 -1.1804146080 -0.4281224930  0.462169200
## 1123  0.4978236405  5.4739698190  1.323570468
## 1124 -0.7993476600  2.2183211438 -1.691121909
## 1125 -3.9988035945 -0.3666333185  0.971408899
## 1126  2.9757793043  1.2103800263 -1.207905652
## 1127  3.2168455611  1.3560833417  1.640610681
## 1128 -4.5117498983 -0.6747293173  1.447160071
## 1129  4.9101798495 -0.8570325356  0.422346972
## 1130 -0.1400434369 -1.1342812463 -1.953725219
## 1131 -3.9879296924  1.1295441836 -1.951571191
## 1132  0.3854116131  3.1617710179  1.239675548
## 1133  2.5025861609 -1.0636855824  1.694932897
## 1134  2.5316768756 -2.0136917666 -1.037866741
## 1135  3.0985128184  0.5560243133  0.405701132
## 1136 -0.4485940462 -0.5428457338 -0.432239181
## 1137  0.4978236405  5.4739698190  1.323570468
## 1138 -0.4411015891  1.4410696209 -0.275910809
## 1139  3.2078751145 -1.1438718648  1.378623041
## 1140 -2.6184161916  0.9136417684 -1.694977703
## 1141  1.3420854206  0.4079987252  0.219301768
## 1142 -1.5933322377 -1.8702849125  0.499597670
## 1143 -2.2503044041  1.3822834135  1.344511500
## 1144  3.2653178221 -1.0927513821 -0.222707944
## 1145  3.2598873950 -2.2475029198 -0.362787718
## 1146 -4.2420622328 -3.1470149131  1.281126153
## 1147 -3.5222022565 -2.4562422675  1.520434304
## 1148  3.1118922152 -0.1504189613  0.768913346
## 1149 -0.4917489810  2.0731795943 -1.762339300
## 1150 -2.7587670823 -0.9095315603  2.034496890
## 1151 -1.0329345734  0.2835064693 -0.926537673
## 1152 -4.5518356660  0.7560945187 -0.890743825
## 1153  1.1616227927 -0.8232016897  0.571486966
## 1154 -2.5577980840  0.1438711806  0.662242944
## 1155  4.5271358444 -2.2264463511 -0.428281329
## 1156  0.4629706566  1.4714045784 -0.042248896
## 1157 -1.0030353110  0.1028975970 -0.418683342
## 1158  0.2650445542  2.4901351758 -2.169249029
## 1159 -2.3820037099  0.2760923473  1.729147097
## 1160 -4.5357593210 -1.4331448878 -0.731040387
## 1161 -3.7374537122 -1.6019833370  0.658463736
## 1162 -2.5771881782 -0.2267333337 -0.310992291
## 1163  1.1445535263  1.4501357000 -3.248805752
## 1164 -2.0095857948  1.0966543506 -0.640950076
## 1165 -3.2143979072 -1.0840257574 -0.245059584
## 1166  3.9347511610 -0.7876504586  0.490008077
## 1167  2.3564993390  1.7968120311 -0.472633421
## 1168 -1.5759541212 -1.7531614549 -0.590556266
## 1169 -0.7064079481 -0.7796690514 -0.932124322
## 1170 -1.8358195685  0.8267655600 -0.378048944
## 1171  1.9656284111  0.9486870018 -0.113828942
## 1172 -0.4849797017 -1.3105253954  1.190971364
## 1173  1.1117128427  1.1251825202 -0.892080701
## 1174  2.4802342662 -1.5209396164 -1.784568080
## 1175  3.4670361391  0.3252783021  2.132810281
## 1176 -0.8206234579  0.2136618328  0.500255976
## 1177  2.3876586333  1.0023167566  1.371836097
## 1178 -0.8664480318  0.0468249505 -1.129730287
## 1179 -3.3644535448 -0.7812331414  1.544980702
## 1180  1.9294358036 -1.3271242219 -1.852988368
## 1181  1.0325584080  2.0422080772 -0.374921777
## 1182  3.8320679761 -0.8615875584 -2.189351919
## 1183 -3.2313405512 -2.0238837273  1.488294046
## 1184 -3.5001493232  0.0180344297 -2.854740117
## 1185 -1.9625075627  0.1562646498  0.163379455
## 1186 -0.5579668045  0.8590518931 -0.245593543
## 1187  2.3466225230 -1.0687756669 -0.073610554
## 1188  3.9103762485  0.0945201070  0.050137026
## 1189  3.7822291393 -0.9565922737 -0.124558516
## 1190 -0.3866841060  1.8353010141  1.414958638
## 1191 -1.3703497740  0.4511122727  0.679043312
## 1192  1.6672112303 -0.7510454801  0.046216839
## 1193 -3.0815287037  2.9857073616 -0.795233569
## 1194  3.1414011800 -0.9328825799 -0.987585387
## 1195 -2.3108797796  0.0597682590  0.039175016
## 1196 -2.7623389606 -0.5001824371  3.489086563
## 1197 -3.9085657063 -1.1069278257  0.841326404
## 1198  3.3744520862 -1.2581221614  0.562600912
## 1199 -2.8863060304  0.3346363530  0.442539840
## 1200 -2.4852861062  1.2441454073 -1.423380095
## 1201  2.1893343105  0.5392034521 -1.800580237
## 1202  2.1568997459  1.0326274758 -1.202146030
## 1203 -2.3540578957 -0.6785859374  1.371803533
## 1204 -0.0875528397  1.1146290455 -0.128414827
## 1205 -3.5550399400  1.0533447877 -1.994315694
## 1206 -1.7327843583  1.7511970215  0.607102930
## 1207 -3.3344792065 -0.8570291684 -1.021775484
## 1208 -2.0990583136 -1.0489835271 -0.039829911
## 1209 -3.0314123755 -0.5360920966  1.008749274
## 1210 -3.5126185209 -0.7714324406 -1.736581482
## 1211  1.6828974288  0.8415877641  0.021484390
## 1212 -1.7414797537 -0.6866398895 -2.322982254
## 1213  4.3920196220 -1.0244969653 -0.421082144
## 1214 -3.6315701293 -1.2579600073  1.051088013
## 1215 -1.7779020933  1.1341290952 -0.452674309
## 1216 -4.0434181918 -2.3416243573  1.061509651
## 1217 -2.7884664274  0.9490122643 -1.869613042
## 1218  3.7155254197 -1.5331111898 -0.732378963
## 1219  1.5811201964 -1.6278680346 -2.016283696
## 1220 -0.2241050116  1.8058798635  1.094090529
## 1221 -6.4212269873  3.5548470483  2.890671379
## 1222 -2.0665703120 -0.0247237906  3.912450915
## 1223 -4.2010341108 -0.6066840700 -0.691621641
## 1224 -5.1058081672 -0.7982826865 -1.668180715
## 1225  1.6068484259  0.9633318429 -0.682313685
## 1226  2.9791587722 -0.5882030086 -1.479525284
## 1227  2.6812970973  1.6269846444 -0.482033043
## 1228  4.4768645940  1.3012865097 -0.910376734
## 1229  2.2299391659 -0.1508513251  0.878520360
## 1230  2.3634324361  0.6864308006  0.384292139
## 1231 -0.2441539067  2.7754242362 -0.460506008
## 1232  0.9410343373  1.0813610867  1.003551706
## 1233  2.5204890068 -1.3833127554  0.297725590
## 1234 -4.3282141548 -0.8211872309 -1.037694909
## 1235  5.0449301550 -1.4268981410  0.329486372
## 1236 -3.0763759055 -0.3210220547 -1.657175133
## 1237  3.5033830864 -0.6811549433 -1.145256529
## 1238  4.4772570608 -0.8208885111 -0.226465936
## 1239 -3.8261378893  1.1010571846 -0.273377521
## 1240 -2.7815326330  0.3758696150 -1.399759734
## 1241  4.7265228496 -2.3933075883 -0.199418388
## 1242  1.1169382769  5.1509809129  0.502422805
## 1243 -2.3322192546 -0.1173088418  1.197151973
## 1244 -1.8445445821  0.6330634210 -2.522935138
## 1245  3.3246182070 -1.4676216033 -0.425197899
## 1246  1.9900334626 -0.3076617672 -0.973199373
## 1247  2.4673126561  0.2700320000 -0.533773000
## 1248 -0.3341844367  1.1680311998  1.481911214
## 1249 -0.0199747106  4.0951594898  0.179904616
## 1250 -2.2538468442  0.2108615954  1.300032329
## 1251  1.7112935234  1.2079768959  0.471903990
## 1252 -3.2663644689 -1.0962996713 -0.353765569
## 1253 -2.4968928459  0.8072930183  0.308405967
## 1254  3.6773434285 -0.9805755153  1.123321501
## 1255 -0.4770207615  1.2205818912  0.324554688
## 1256  4.6644911654 -1.2960944137  1.271110122
## 1257  3.9989460759  0.2904423054  2.312898916
## 1258 -3.8807012040  1.3262393724 -1.038844738
## 1259  1.2342823992  0.5082177583  0.384873277
## 1260  0.5664593152 -0.5131891725  2.810910180
## 1261 -3.1762859324 -0.8708617489  0.111203316
## 1262 -0.4672058466  1.9980772520 -1.268641991
## 1263 -2.1821772367 -0.6439270582  0.737893677
## 1264  5.3643047553 -1.0224595352 -0.561052719
## 1265  2.6518516538  0.1436106517  2.108825057
## 1266 -4.1160980381 -1.5208909960  0.073316234
## 1267 -4.1187611118 -0.6191424066  0.345801397
## 1268 -2.0859515897 -1.3385200433  0.207726354
## 1269 -4.4413684436 -0.1646891072 -1.041511214
## 1270 -3.4472494557  0.0285259145 -3.063335794
## 1271 -0.2597886425 -0.7592199271 -1.676111088
## 1272 -3.3176675354 -0.0083788206 -0.407761799
## 1273  1.9841336972  1.5205085749  0.547365596
## 1274  4.2483456897 -1.8412091621 -1.054754197
## 1275 -3.1169020056  0.4348889332  2.032882558
## 1276  4.9862283727 -1.2286810126 -1.023792813
## 1277 -3.0481462767 -2.3930388488  1.411989815
## 1278 -0.9679617231  2.0985566850  0.981860676
## 1279  3.7388273624 -1.0700296542  1.444880255
## 1280  1.1138988237  2.4842147963  0.333951326
## 1281 -0.3857904736  2.4883326108  1.307249691
## 1282  3.5949775670 -0.9175590777 -2.599069750
## 1283  1.8631256990  0.5250019315 -1.788821608
## 1284  2.7718660740  2.6977700796  1.495023417
## 1285  1.3507755605  0.2341833109 -2.362596571
## 1286  2.9792933042  0.4692409531 -0.371819082
## 1287 -0.4392922927  1.0865484295 -0.148747879
## 1288  3.2747883579 -0.0297896283  0.764369731
## 1289  0.3220171992 -1.5915928654  1.377239598
## 1290 -3.8659524756  1.7017841909 -1.357517632
## 1291  1.0575256777  0.1918382682  0.963094152
## 1292 -3.2607816602 -1.0399969079  0.128351236
## 1293 -2.4047740205  0.4746727637  0.444526533
## 1294 -2.4904390158 -1.9334711753  0.068108409
## 1295  4.5749901575 -0.5756047130 -0.700471967
## 1296  0.3965931930  1.0528102241  0.020430466
## 1297  4.0614908941 -0.9823600597 -0.455376275
## 1298 -0.1602322187  2.3614525554  1.770818949
## 1299 -3.2647711352 -1.0895153756  0.921503811
## 1300  2.2114899178  2.8339125402 -0.312318273
## 1301 -2.4916636051 -1.8614122668  0.629014493
## 1302 -1.2598153047 -0.1544383646 -1.039788454
## 1303 -2.4556160466  0.1551091873  2.200675184
## 1304  3.5564911743  0.0848889961 -1.321623202
## 1305 -2.1799379838 -1.1633254611  0.985788169
## 1306 -0.4270353347  1.7677916156  1.379265140
## 1307  3.5192113943  0.4438805720  0.461881453
## 1308  4.9271970315 -1.3259134265 -0.513531250
## 1309  1.9800418169  1.2240016026 -0.310934222
## 1310  0.5584438854 -0.5751661958  1.220147007
## 1311  3.3987709470 -2.4361691641  0.596489527
## 1312 -1.6665182315 -1.3676658993 -0.560335928
## 1313 -3.2008528306 -0.8411918998 -0.650269594
## 1314 -0.1605197790  1.0389916774  2.172484885
## 1315 -0.7996679957 -0.0601864482 -1.097610525
## 1316 -0.0210949975  1.1901694798  0.493228175
## 1317 -0.0232986662  5.3467689516 -0.925018470
## 1318 -2.4728729850 -1.7515693534  1.157444726
## 1319 -4.2532766543 -1.4609810732  1.777043275
## 1320 -1.6061412501 -1.0431182647  1.339941221
## 1321 -3.2965706563 -1.3312658704  0.674305931
## 1322  1.1958287650 -0.7837369260  0.153094497
## 1323  1.3251867706  1.3821377091  0.148743065
## 1324  3.2079194869 -0.8621545114 -2.212954837
## 1325  3.0908561873 -0.3495133506 -2.284559728
## 1326  2.8613231632 -1.5237496460  0.721592771
## 1327 -3.9162005612 -1.1495542413  1.897785587
## 1328  0.0414962046 -1.2488516868 -0.777970846
## 1329  3.6305587229 -0.6236962189  1.091401644
## 1330  2.9077602300  0.2358538245  0.798026527
## 1331 -1.9185925948 -0.4273174007  1.221296376
## 1332  2.3743335239 -0.5492763451 -1.767225879
## 1333  3.9263432325 -1.5279000831  0.018391177
## 1334 -2.1489226976  0.6973931276 -0.339743103
## 1335  0.5909890805  3.0335309168  1.477352884
## 1336 -2.7575343806 -1.1680367570  2.301034031
## 1337  0.2569625014  2.7605047938  1.564413000
## 1338  3.1679225168  0.1767951481  1.246312046
## 1339 -0.1134749102  0.7573760983  0.738236968
## 1340  2.9918874634  0.1631723957  0.075262723
## 1341 -0.0323901571  0.6198715055  0.507488789
## 1342 -3.3903695214 -0.9473080909  0.631712418
## 1343 -0.9911057861  0.2369561490  0.107391889
## 1344  0.1463622001 -1.1667487701 -1.374529071
## 1345 -0.4134374072 -0.7054689792 -0.124085762
## 1346 -2.2938522265 -0.1077803531  0.140822716
## 1347 -4.4747146219  1.1409011039 -0.987677723
## 1348 -0.5033554748  2.5715527741 -1.308994451
## 1349  1.9027978512  2.3986134064 -0.484083389
## 1350 -0.3722753177  1.9615349756  0.854561056
## 1351 -1.1309550379  2.3319811389 -0.259042421
## 1352 -2.5270117272 -0.9729507212  0.972331098
## 1353 -1.3016382426  1.4321024968 -1.115335369
## 1354 -0.6813629058 -0.1523952276 -0.988931525
## 1355 -0.8938751619  0.8160297742  0.863855819
## 1356 -3.4737831758 -0.0520341607  1.031563898
## 1357  4.4211126531 -1.5628459052 -2.016011307
## 1358 -4.2977532620 -0.4600611910  0.706460152
## 1359 -3.6722496664  0.0923085302  0.543699148
## 1360  3.7412131896 -1.4384707415  0.387075035
## 1361 -0.4447610408  1.8920835876 -0.101306833
## 1362 -1.8775230362 -0.6161199190  0.879612530
## 1363 -3.6301836470 -0.5878052775 -2.067263928
## 1364 -2.2670254697 -0.6856433835  1.731263446
## 1365 -4.3057249091 -0.6453813638 -2.357319872
## 1366  1.0089649773  1.6169397068 -1.559153379
## 1367 -4.7400849709 -0.9844160848 -1.033171804
## 1368 -2.6908232426 -1.2057469080 -0.137340834
## 1369  1.9656284111  0.9486870018 -0.113828942
## 1370  0.1946836825  1.6486247301  0.337242132
## 1371 -0.6483302321  2.0454163518  0.541121414
## 1372  1.0088800202  0.3994799649 -0.688665685
## 1373 -1.4781978272  0.2200352610 -0.674182142
## 1374 -0.4597909398  1.5393284344  0.919612355
## 1375  1.1198064502  2.6752334821  1.044793135
## 1376 -2.2102615558  2.3927589832  0.093313651
## 1377 -0.4822743000  1.7436935307 -0.654040467
## 1378  2.6926819945  0.4418373686 -0.486606997
## 1379  1.0277429411  0.7267309161 -0.058376937
## 1380 -2.0911529613  2.5695840722 -0.583497358
## 1381  2.2669470032  0.5848108482 -0.394127177
## 1382  2.4277530530  0.5174455324 -0.394109486
## 1383 -4.0784335314 -0.6678108972 -1.134937160
## 1384  4.3385986636 -1.7334414697 -1.623487635
## 1385 -1.0945391059  0.2099615570  2.212424748
## 1386  1.5510916574  0.0502622770  1.263946449
## 1387 -2.1567344862 -1.6242278397  2.024837983
## 1388  0.5215200032 -1.9777238745 -0.790085237
## 1389 -2.9965588645 -0.0019112825 -0.108306737
## 1390 -3.6029197380 -0.1936489978 -0.551791513
## 1391 -5.5619120015 -0.1284946554 -2.519820145
## 1392  3.2741367687 -2.1177904096  0.801034174
## 1393  2.1464351158  1.8376138915  1.053607502
## 1394 -3.4391192678 -0.5199157580 -2.233925598
## 1395 -2.7233563842 -0.8765589489  1.384920072
## 1396 -2.2199252738 -0.5458341444  1.231533846
## 1397 -4.3465940256 -0.1755092230 -1.104540136
## 1398 -4.2348008501 -0.3359138322 -1.386514267
## 1399 -2.5814619332 -0.6413668524  0.498907788
## 1400  0.8233313645 -1.0234640183  2.004033164
## 1401 -3.6845294216 -0.6719266897 -0.061004661
## 1402 -3.2033288505 -2.0463206894  0.723158210
## 1403  4.0016828283 -1.4724085808  0.260185034
## 1404 -3.4073819184 -0.3640855288  0.234793840
## 1405 -0.2606822838  1.4095383347  0.338819703
## 1406 -3.4135141413 -0.1748684373  1.163833121
## 1407 -0.6483302321  2.0454163518  0.541121414
## 1408 -3.4559545727 -2.2687153697  1.710707473
## 1409 -0.8396558548 -2.5623084667 -0.122931390
## 1410 -1.4696648534 -0.2692670117  1.346170352
## 1411 -2.6054839775  0.7957082090  0.144809788
## 1412 -3.8683547496 -0.1043372734 -1.362768546
## 1413 -2.7932464078  0.6109897169  1.758502739
## 1414  4.3521010238  0.3464937038 -0.559096171
## 1415  5.0871869484 -0.3648267873 -0.073346784
## 1416  1.0580037663 -1.9466491293 -0.291319049
## 1417  3.3502429302 -1.3976589306  1.336712905
## 1418  1.2854885548  2.7135306220  0.564703282
## 1419  3.2390065310 -1.1394687250 -0.280414060
## 1420  1.5988068481 -0.6789287299  0.321769521
## 1421 -0.4548965609 -0.2547913031  0.699692478
## 1422  3.3074450486 -2.2338385764 -0.501337169
## 1423  3.7078670776 -1.2075931809 -1.238376595
## 1424 -3.3345690234  0.0970214201 -1.645453196
## 1425  1.0935812719  0.9577417979  0.764087128
## 1426  2.9284060818 -1.9465699955 -0.257782239
## 1427 -1.3867953186  1.0931005256  1.067940305
## 1428 -2.3786346593  0.3900313923  1.091176827
## 1429  5.4121038038 -1.8302785238 -1.293098063
## 1430  1.7101045979 -0.0590776577  0.719237828
## 1431  0.5362663881  1.1925218033 -2.040338420
## 1432  1.9331219949  1.1325376059  0.338132597
## 1433 -3.6579387406  0.3237943345 -3.033220721
## 1434 -3.1646769898 -0.9864678513  1.002399606
## 1435 -3.6641535483 -0.6575588329 -0.290895044
## 1436  2.4543966868 -0.1017736329 -0.092952806
## 1437  0.3262017057  2.2226960412  0.121187970
## 1438  4.0759771216 -0.9141829089 -0.622136294
## 1439 -4.4217560022 -0.6371031230 -3.423112209
## 1440  3.1336873364  1.0222650805  0.983548901
## 1441 -2.8153931778 -0.3095800329  1.561667183
## 1442 -1.4677822901  0.6897016075  1.230624178
## 1443 -3.5099331933 -0.8570391254 -0.547392386
## 1444  2.5624646057  1.7378569409 -0.567700561
## 1445  2.7429976186  0.7283566537  0.623985950
## 1446 -1.4098810496  0.0501085768  0.124980729
## 1447  2.9792933042  0.4692409531 -0.371819082
## 1448 -2.6337919448 -0.5589188908 -2.641395947
## 1449 -0.4977176773 -1.4181208092  0.017110760
## 1450  0.8206541271  1.5937653803 -3.066250679
## 1451  3.0263877277  2.1709289726  2.275580078
## 1452  2.1506420791  1.2095372286  0.834981783
## 1453 -3.6716160584 -1.4549195160  1.709511238
## 1454 -3.2033515052 -1.0968504558  1.432666093
## 1455  3.0023504280 -0.2971141313  2.278617916
## 1456  2.4374901643  0.2882454851 -0.085934047
## 1457  2.9518253436 -1.5457354595 -1.161415055
## 1458 -1.3407201927  4.2745295166 -2.960382019
## 1459  4.0592693842  1.2431746105  0.251490245
## 1460  0.1362925702  1.3687928325  0.297911415
## 1461  2.0669555969  3.3388129217  0.052819342
## 1462 -1.5654919787  2.7124069694 -1.548028822
## 1463  5.6351488594 -1.2344339491  0.319657981
## 1464  1.3211718375  1.7460244019  1.258902492
## 1465 -2.4829877999 -0.7036935660  0.941284484
## 1466  3.9549715303 -1.5645446304 -0.156405349
## 1467 -3.3870169565 -0.7666859757 -0.065627613
## 1468  1.1021487761  2.8399452814  0.069391668
## 1469  3.0952491582  1.4562146110  0.926937795
## 1470  3.8746042999 -0.4807630394  0.878102472
## 1471  0.2757522988  2.0602677248  0.647118274
## 1472 -3.2033288505 -2.0463206894  0.723158210
## 1473  0.3347626304  2.2631576117  1.986873769
## 1474  0.7175731018  5.6116629596  1.780977545
## 1475  2.4700772442  1.0496970607 -0.681718077
## 1476  1.5151855941 -1.3007282413 -1.525675883
## 1477  1.2056279311  3.0342054255  0.915613901
## 1478  3.0958745486  1.9271835147  3.401950677
## 1479  3.9768985784 -1.6621654865 -1.062020207
## 1480  1.7419336485  1.6932412686 -0.072024427
## 1481  3.5125749918 -2.5219361424  0.114809581
## 1482 -2.6173497869  2.0134942298  0.366605315
## 1483  3.7157604041  1.6536358473  1.861435367
## 1484  4.1776685265 -2.1257389051  0.335951452
## 1485 -2.2049188725 -1.7536593372  2.101278264
## 1486  0.0954625212 -1.0436305329  0.642202952
## 1487  0.9971445226  3.1091633022  1.044392130
## 1488 -1.9941640369 -1.2412989374  0.615488029
## 1489 -2.1904231735 -0.0006173919 -0.069186965
## 1490 -2.1496552578 -0.0845237731  1.870330884
## 1491  3.9784092166 -0.3605431176 -0.317814625
## 1492  4.1346468239 -1.3344744004  0.892326514
## 1493 -2.0291800083 -1.4750013591  0.056937024
## 1494 -3.0550200251 -0.4412265955 -1.515880049
## 1495 -6.3955477909 -0.7504327591  1.000368478
## 1496  0.7416747048  4.6514875476  1.676660527
## 1497  3.9873138038 -1.8513414954 -1.967074898
## 1498 -2.2394750144  1.1160205592  0.888887342
## 1499  3.5476560010 -1.7614763571 -1.234460643
## 1500 -2.6478247241 -0.9656096008  0.621706548
## 1501  2.2771999740  1.7938998537  2.098972200
## 1502 -1.1533452301  2.2241175682  0.371971815
## 1503  1.6821385241  1.1778545764 -1.671826913
## 1504 -2.4734534071  2.0022890933  0.227518847
## 1505 -2.5306132869 -0.4580217301  3.139310681
## 1506 -0.8302403309  1.1931623019  3.354900149
## 1507 -4.4637066903 -1.0953961012  0.954856749
## 1508  3.3674726069 -1.1896741535 -1.397257884
## 1509 -0.0494467370  2.2033101684 -1.607586507
## 1510  2.3936510109  2.0175369718  0.108047596
## 1511 -2.6166916812 -0.9384637457  0.295211341
## 1512  2.6975763752  2.1219254549 -0.134050419
## 1513  3.5707062046 -1.0395732654  1.660672607
## 1514 -4.4695512927  0.5054373726 -2.651947521
## 1515  0.8922471629  2.1506994932  0.679581832
## 1516 -2.4661358234 -0.2143565583  0.095597834
## 1517 -2.1021335373 -2.1022640572  1.884090656
## 1518 -1.1226676361 -0.0874260079  1.158967894
## 1519  2.1015926084  0.7016916536  0.776398906
## 1520 -4.0679414570 -0.9815917885  0.584276772
## 1521 -1.4847372250 -1.2039787693 -2.082086068
## 1522  2.6149992392 -2.1992267214  0.040596173
## 1523  2.4332561355  0.2232344503  0.742403262
## 1524  3.0106222261 -1.6952817146 -0.406909355
## 1525  3.0963401858  1.7914210562  1.462965973
## 1526 -4.0048177502  0.2983712764 -0.665655704
## 1527 -1.0183066660  3.1529725647  0.235384733
## 1528 -4.0764095387 -1.2974701241  0.550840089
## 1529 -2.1871782395  1.1708243252  0.619104576
## 1530  3.1658438485  0.9052811622 -0.013271522
## 1531 -1.4696648534 -0.2692670117  1.346170352
## 1532 -1.3925978009  3.1242864023 -0.475363211
## 1533 -4.4149636947 -1.0955728295  1.249487301
## 1534  3.3782967118 -1.8151709734 -0.712732806
## 1535 -2.9397249448 -0.4516984894 -0.294501870
## 1536  0.3682249880  0.4595846124 -0.360905690
## 1537  1.9364007317  1.5243534417 -0.465191939
## 1538  2.7293093307 -0.4780658605 -1.146170578
## 1539 -0.0528403024  4.6023133341 -0.939412045
## 1540  1.2357074810  4.5736181504 -0.765203464
## 1541  4.3490029545  0.4297254486 -1.743673033
## 1542 -3.4786184759  1.5325691839  0.081402029
## 1543  4.6710322242 -2.2376514875 -0.567367797
## 1544 -3.1385620613 -0.9695712224  0.387720653
## 1545  0.3347626304  2.2631576117  1.986873769
## 1546 -1.9941647728  2.1959902723 -1.541229519
## 1547 -1.4480056777 -0.7473618232 -0.027928801
## 1548  2.5070996742 -0.8704237619 -4.067109282
## 1549 -1.4422926608 -1.4645637963  1.337677885
## 1550  1.1866148607  0.1723388318  0.406113156
## 1551 -4.8575252601 -0.1796498043 -1.601092870
## 1552  2.8144326231 -1.0351595780 -1.757776256
## 1553  2.8240855884 -1.3118702249 -0.659834145
## 1554 -2.1129778331 -1.0867681209 -1.261387173
## 1555  3.8512495022 -2.4383211012 -0.434255529
## 1556 -2.9324795256 -0.3322271912 -1.796261601
## 1557 -0.8273513363  1.5605549894  0.390836054
## 1558  1.1482371864 -2.2620224695 -0.162030615
## 1559 -1.5267804729  1.2465195684  0.790546468
## 1560  4.1577938477 -1.0856370994  0.677356593
## 1561 -2.0095857948  1.0966543506 -0.640950076
## 1562  2.8398264297 -1.5586647177  0.242526101
## 1563 -3.2181265476 -0.9622663699  3.001086039
## 1564 -0.4392922927  1.0865484295 -0.148747879
## 1565 -2.7297862421 -0.6806830134 -2.820913000
## 1566 -0.1470825371  2.6379284773  0.045608397
## 1567 -3.4968086549  1.3018029887 -0.777502632
## 1568 -0.3803431005  0.9119712804  1.513060957
## 1569  3.2735571194 -0.2601403044  1.070250058
## 1570 -2.9628492702 -1.6037114174 -2.114466133
## 1571  0.4191394472  1.8954772507 -0.157939420
## 1572  5.3520738583 -0.5904426489 -0.248285033
## 1573  2.8621083203 -0.3617460773 -1.345301803
## 1574 -3.7105735486 -0.1289148781 -1.084722541
## 1575 -3.5934255077 -0.1443601832 -0.607326409
## 1576 -4.1889236558  0.6690724756 -2.179559823
## 1577  2.4134800263  0.4988930339  0.153803247
## 1578  1.6104179526  1.2703271517  2.621539544
## 1579 -0.5534504942  1.2999233492 -0.912833897
## 1580  0.6942486822 -0.4573685698 -2.814370058
## 1581  3.4617071991 -1.2059709897  1.508197718
## 1582 -2.7226794796 -1.2426506349  2.764674698
## 1583  3.4271772332 -1.1131243968  0.912047936
## 1584 -0.8688646607  0.8091321404  1.154818452
## 1585 -3.8100331023 -0.0513699047  0.276722591
## 1586  2.8299421721 -0.2908171617  1.566318639
## 1587 -4.3718902347 -1.2566015326  0.371907035
## 1588  0.2719205394 -1.3560966150 -0.097660751
## 1589  0.0414962046 -1.2488516868 -0.777970846
## 1590  1.6283339940  3.9725611615  0.616994866
## 1591 -2.2172294148  0.4296601667 -1.893703955
## 1592 -2.6861775904 -0.1866717193  1.294549648
## 1593  3.4752605404 -1.5778752397  2.221775016
## 1594  0.1150663479  4.0624506828  1.279835696
## 1595 -0.4502298254  0.4468645304  0.389213531
## 1596 -1.9388085532  1.0200716103 -2.742610718
## 1597  1.8604616652  2.0814379398  1.382846757
## 1598  1.9465943320 -0.3636119247  0.154384912
## 1599 -3.0027758671 -0.3760580168  0.173771601
## 1600 -2.1877542682 -1.0429970189  1.430779367
## 1601  3.1000045879  1.2766144138 -1.707122879
## 1602 -0.8823502153  0.8303697790 -0.983423430
## 1603 -3.0167685054  0.4588660500 -3.231792322
## 1604 -4.7225940819 -0.9626452024  0.200258543
## 1605  0.5691920147  1.0907242176 -1.386950495
## 1606 -0.0586076697  3.9942675325 -1.611333934
## 1607  1.0188274644  1.1433839926  0.297312266
## 1608  3.6352382217 -1.2725290137 -0.555100644
## 1609 -3.4023960958 -0.4230472107 -1.068095928
## 1610 -3.4342495676 -0.9184879674 -0.058819942
## 1611 -3.5261305848 -0.7798241507 -1.301916808
## 1612  1.1792801254  2.3604581153 -1.956867290
## 1613  1.0616275984  3.5381880994  2.788820656
## 1614 -3.6299149037 -1.8375179393  1.341828645
## 1615  2.4369253143 -1.1015466813  0.679027088
## 1616 -3.3501017678 -1.7072096125 -1.281151353
## 1617 -0.2444234745  0.7902962093 -0.624330929
## 1618  2.5746800365 -1.5601697046 -0.190251225
## 1619  2.9449030028 -0.4393497394  0.791362562
## 1620 -1.2598392589  0.4702164947 -2.288382322
## 1621 -2.7547778705 -1.5462963152  0.485101414
## 1622  3.6550141595 -0.3943387733  0.488947815
## 1623 -2.0586983890  1.3803632740  1.057591088
## 1624 -3.7885598327  0.3809834458 -1.931895269
## 1625 -2.7407589885 -1.0398115349 -0.651477010
## 1626 -0.4626260659  1.2706847174 -0.537360629
## 1627  2.5373708210 -0.1816905878 -1.913780702
## 1628  1.6214147611  1.3071989394  2.070676883
## 1629  3.7024798533 -2.1151016814 -0.462631972
## 1630  0.6306624220  3.5279754474 -0.166608088
## 1631 -1.2648103045  1.5559845998  1.349588072
## 1632  2.4995918962 -0.3259489816  0.034556700
## 1633 -2.3470117020 -0.1134349480 -0.067841191
## 1634 -2.3208013838 -1.6144215306 -0.753648863
## 1635 -2.1890585892 -1.4111435277  1.662877324
## 1636  1.6245483237  0.7532219565 -1.429581729
## 1637 -2.6049773554  0.0587497261 -1.573147220
## 1638  2.7834426184  1.3260677193 -0.378532140
## 1639  4.4768645940  1.3012865097 -0.910376734
## 1640 -0.6068353222 -0.0602252014 -0.945730195
## 1641  3.9579435187 -1.1421351958 -0.235922639
## 1642  3.8823659684 -0.3757030120  0.130231896
## 1643  3.5066626323 -1.0414142230  0.401464558
## 1644  4.4137139628  1.3916864899  1.657686364
## 1645 -1.7415107411 -2.3773645467  2.471167941
## 1646 -0.6881718839  0.5137713063  1.657943794
## 1647 -3.7384318423 -1.6081223121  1.923131975
## 1648 -1.0533436384  1.4015620299 -1.502290600
## 1649 -2.6541965045 -1.3109368021  0.048357568
## 1650  4.2520035653 -1.4606830308  0.662539387
## 1651 -1.7111624039  0.2431000937  0.212705053
## 1652 -3.7021861780  1.8217819990 -1.299254907
## 1653 -4.6987974120 -1.7911081862 -2.037626222
## 1654 -0.3609760849  0.4544809193 -0.368482099
## 1655 -3.7364048697 -0.7435676488 -1.155107492
## 1656  3.6099671218 -1.0366825973  0.282160467
## 1657  1.1920677479  2.1977262484 -0.062696423
## 1658  2.8020317938  1.0668793277  1.254335567
## 1659  3.9101037716 -2.1161835832  0.755552547
## 1660  0.5680134957  1.8055747735 -0.181449539
## 1661  3.0358727830 -0.6513032443  1.002566459
## 1662 -4.5854939997 -0.4057886609 -0.552196906
## 1663 -3.6865410023 -1.4281655525  2.454401780
## 1664 -2.4289872399 -1.2263634795  2.319925682
## 1665  2.4075079892 -0.4521704609 -0.290710455
## 1666 -3.9520185766 -0.1915422057  1.141195138
## 1667  2.0769082465 -1.6436549535 -1.362564825
## 1668 -4.2432500343  0.8378936751 -1.753295440
## 1669  3.7884617735  0.9383646902 -0.559171680
## 1670  2.5328163337  2.0720843940  1.830187728
## 1671 -3.5550399400  1.0533447877 -1.994315694
## 1672 -2.7447188136 -0.7629312820  1.645977514
## 1673 -0.3206438026  2.4599770090 -2.042822379
## 1674 -2.0046450083  0.2498037252  2.900254727
## 1675 -1.8512218718  0.3857366414 -0.588051851
## 1676  2.7617019524  2.1195833180 -0.465272017
## 1677 -0.3691135468  0.7417685631  0.821663494
## 1678  1.7307109925 -0.4652183742  0.204539267
## 1679 -0.3384423443 -0.9201488177 -1.923063366
## 1680  4.8581237357 -2.0256966504 -0.918505702
## 1681  3.5087865476  0.6254110060 -1.653555072
## 1682 -2.5156659729 -1.0923625656  0.148766346
## 1683  4.3885063486 -0.7764023061  0.499923050
## 1684 -2.6687219142 -1.9372523689  1.987323900
## 1685  0.5012142714  1.7069320883  3.185780453
## 1686  2.1373838834  0.2795157868  0.361276104
## 1687 -3.7496508324 -0.6744193543  0.269378945
## 1688 -1.0616795459  1.3971194023  1.075157084
## 1689  2.6385097276  1.7579685242  0.344935802
## 1690 -4.8086754648 -1.3136903268  0.844248896
## 1691  4.5323995616 -0.4126499685 -1.600632730
## 1692  4.8561938141 -1.8614168759  0.479393293
## 1693  4.4345771189 -1.0853363379 -2.087511894
## 1694 -0.8175271207  1.4450959366 -0.770326281
## 1695 -2.4445395422 -1.0161175689 -0.040409773
## 1696 -2.1340191471  2.6242746862 -0.560818412
## 1697 -2.7923709622 -1.2851288140  1.088678377
## 1698  2.6975763752  2.1219254549 -0.134050419
## 1699  3.8977273059  0.5412707054  0.238247186
## 1700 -3.3365819753  0.5697202846 -2.489219287
## 1701 -2.7233563842 -0.8765589489  1.384920072
## 1702 -3.1970433753  0.1503402029 -0.484949075
## 1703  4.3938673386 -2.0470282792 -0.488513691
## 1704  2.1424015438 -0.7810769822  0.659847929
## 1705  2.3306192110  1.2186402214 -0.567201747
## 1706  1.2956374804  0.5782397003 -0.340909847
## 1707  3.5871280774 -1.1979728260 -0.691046015
## 1708  0.1019621269  0.7253891487 -0.478374681
## 1709 -3.3442931546 -0.2282524811 -1.214778354
## 1710 -1.5243783884  0.0140650549 -0.921617911
## 1711 -0.1825030907  1.4774977017  1.431751444
## 1712 -0.8352400091 -0.4715016052 -1.812790164
## 1713  3.9576917599 -0.1884396300 -0.336750870
## 1714  0.2243663287  2.2818484168 -1.338698915
## 1715  4.3734757756 -0.9259464165  0.301925691
## 1716 -3.2290319812 -1.3208360022  1.326093111
## 1717  3.3696008563 -0.1029516327 -1.206057267
## 1718  2.8205899085  0.4819781234  1.174733105
## 1719  3.9765282424 -0.3435076788 -1.887365788
## 1720  1.7397679513  1.9620931634 -0.772153428
## 1721 -0.0535973950  1.4430179566  1.716292069
## 1722 -3.0042399858 -2.1249124402  1.144403353
## 1723 -0.2291687211  2.6489206835 -1.852523360
## 1724 -3.2993823020  0.6631578075  1.002082459
## 1725 -3.1176881942 -0.9473655187  1.574681029
## 1726 -2.4570217146  0.4814364317  0.304997019
## 1727 -1.5325861533  0.6645022139 -0.089083910
## 1728 -3.2999070039  0.0697966422 -2.652268420
## 1729  2.4289727043  0.1702584499  1.086971206
## 1730  2.2576568716 -0.1059171762  1.644775455
## 1731 -3.3193223879 -0.8415456667  1.644301142
## 1732  1.3473478682 -0.8789825336  0.445248131
## 1733  3.0747810402  0.0875499577  0.581832070
## 1734 -2.7999331688 -1.4205629353  2.016811533
## 1735  2.9622286281  1.2401689510  0.695808780
## 1736 -4.2073814439 -1.2943858022 -2.073258514
## 1737  3.3987709470 -2.4361691641  0.596489527
## 1738 -2.7704369342 -2.2869429481  0.708948897
## 1739 -1.4194288069  0.5088481086 -0.451475135
## 1740 -2.0432118083  0.9752979927  0.449334659
## 1741 -0.3452093894  0.0631030607 -1.471900494
## 1742  2.3677841467 -1.0396860943  0.434178212
## 1743  3.7753360944 -1.4219709721 -1.730077422
## 1744 -4.2851414488 -1.2197494527 -1.523105877
## 1745 -2.1340191471  2.6242746862 -0.560818412
## 1746 -1.5335110931  0.7197432338  2.065867208
## 1747 -1.3402237044  0.9639876510 -0.069594547
## 1748 -2.8164204370  0.3527894326 -1.416789428
## 1749  2.2914689018 -2.5966005339  1.200465677
## 1750 -3.0581811481 -1.2071572105  2.669916280
## 1751 -2.1578090439  0.5205844435 -1.553621332
## 1752  3.2119585730  0.2763142760  1.341842311
## 1753  2.2013249093  1.4041457845  0.240109799
## 1754 -0.6802897109  1.7535125026  0.800863929
## 1755 -1.2308709060  2.1144236216  0.980233759
## 1756  0.1167488352  1.8683844333  1.110495008
## 1757 -2.1019525843 -0.7096221049  0.617857263
## 1758  2.0225970149  3.8161366218  3.854220838
## 1759  3.2665328169 -0.5877542262  0.473173948
## 1760 -2.0530665573  0.2715250595  0.005046854
## 1761 -3.4472494557  0.0285259145 -3.063335794
## 1762  4.4401947386 -1.5452494896  0.041832211
## 1763 -2.0291800083 -1.4750013591  0.056937024
## 1764 -1.1378582615  1.8074950436 -0.032951458
## 1765 -3.6029304355  0.1085143406 -0.857029367
## 1766  0.9315941604 -0.5575330562 -0.476938619
## 1767 -2.3872954353 -0.6590348872 -0.566794883
## 1768  1.1301229169  0.1825733047 -0.667943077
## 1769  2.4369253143 -1.1015466813  0.679027088
## 1770  3.7065411106 -0.5321792816  0.308573025
## 1771  3.4469507811  1.2871857461  0.613302229
## 1772 -1.7375909786  1.1581228282 -1.525350104
## 1773 -0.0444893781  3.1089027943  1.025643900
## 1774 -3.7351774253  0.1767763269 -1.479984125
## 1775  1.7419336485  1.6932412686 -0.072024427
## 1776  0.5252605212  2.1792646351 -0.461489824
## 1777 -3.1023667424  0.2073952435  0.747308508
## 1778  3.6498893076 -1.9593035861 -1.381976779
## 1779  3.4205988142 -0.6875049063 -0.168527319
## 1780 -3.9549411663  0.7060055425  0.124679493
## 1781  1.1544176803  2.9363427528  0.578111599
## 1782  2.3188894195 -1.6470516148 -1.669500431
## 1783  2.7490609010 -1.6068187093  1.959320453
## 1784  3.5247157418 -0.0861608496  0.350722066
## 1785 -4.5136320753 -0.1833126666 -0.820988988
## 1786  2.3530864355 -0.0957490207  1.721398473
## 1787  3.7537756402 -0.2900873745  0.398769424
## 1788  0.6305293379  1.3861835171  0.029224511
## 1789 -5.1147961957 -0.2495247710 -2.460927974
## 1790  2.0780594521 -0.1586682189 -0.100614545
## 1791 -3.5872033575  0.6494535930 -1.129561928
## 1792 -2.1489226976  0.6973931276 -0.339743103
## 1793  2.9828913057 -0.1955742765  0.650862230
## 1794  3.2212065055  0.2707700769  3.074438597
## 1795 -3.7353747220 -0.5194968460  0.148490087
## 1796  0.1881430008  0.7286022534 -1.013151232
## 1797  3.4508140955 -1.7947533193  0.732780963
## 1798  3.8235859179 -0.8716075634 -1.477869721
## 1799 -1.1893185967  0.2450236536  1.372831233
## 1800 -2.1396167077 -0.3302531381  1.563105761
## 1801 -1.6457357742  1.0175460897 -0.284003469
## 1802  2.0995960209  2.9132938847 -0.308047555
## 1803 -0.8489064444 -0.0240774067  0.255497221
## 1804 -0.6767270781  0.2024566964  0.361169508
## 1805 -2.5240344628  0.4874017102  0.489257642
## 1806  4.1376048285 -1.5214516977 -1.165350357
## 1807 -0.7036754008  0.8384882775 -0.437442445
## 1808  0.7971379574  1.0925662232  1.142638174
## 1809  3.3157600747  4.4400513684  0.722446465
## 1810 -1.9972175831  0.9537760748 -0.105220709
## 1811  4.1411056016 -1.4905535922 -1.358629943
## 1812 -3.4356840252 -0.7950471287 -0.964186518
## 1813  2.9395631524 -1.0169201687 -0.438901362
## 1814 -0.0009284632  2.1848089456 -0.387922135
## 1815  2.9292371286  0.1042052632 -1.337406910
## 1816 -6.8295483456  5.2541656033  0.914713136
## 1817 -2.3677740139 -1.9360108271  1.599834167
## 1818 -0.0836109629  1.4384110939 -1.796251933
## 1819 -3.6323547902 -1.5181369498  1.004010068
## 1820  4.4507288421 -0.9977855100  1.867176640
## 1821 -4.5586570858 -1.1702580912  0.620456982
## 1822  3.5253477185 -0.6755448131 -0.200203459
## 1823  2.3050846891 -0.9107022586 -0.611613356
## 1824  3.4210403939 -0.5016256152 -0.523616560
## 1825  1.8242036198  2.2218007322 -1.464171278
## 1826 -2.1971618159  1.3123630084 -0.806559313
## 1827 -2.1813333002 -0.9469979069  0.381322339
## 1828  3.9065176023 -1.3639673092 -1.188239287
## 1829  1.2772779710  0.8000935043  2.165398425
## 1830  1.7933043961  0.2689325026 -0.747821089
## 1831 -4.6163880763 -2.3966106119  1.374670959
## 1832  1.3222307112 -1.0779911155  0.822742099
## 1833  2.5909413951 -1.0957768283  0.569550838
## 1834  3.6305587229 -0.6236962189  1.091401644
## 1835  0.9044863992  4.4319408913  1.503479215
## 1836 -2.9707840605  0.4992383431 -0.581885284
## 1837 -3.9503002154 -1.9268869356 -0.161576325
## 1838 -1.1714982866  2.7426029501  0.923186584
## 1839  5.2612354365 -2.1795872162  0.158967042
## 1840 -3.0794794627 -0.5106367165 -1.380571406
## 1841  3.6153708784 -0.1326167688  0.828440038
## 1842 -3.4949310572 -0.2006208492 -0.775351347
## 1843  0.9640350298  2.7708783198 -1.601888598
## 1844 -2.7138029067 -2.0576660305 -1.692317576
## 1845 -3.0972394599 -2.7783524587  1.594057571
## 1846  2.3411287130  1.3194595161  2.017117220
## 1847  4.4374639348 -2.0496324221 -0.356209063
## 1848  2.8976076704  1.1003675237 -1.694415157
## 1849  3.0362256513  0.7598874520  1.142877211
## 1850  3.2877104278 -0.2984176510  0.376874586
## 1851 -3.4957750098 -0.3113573064 -1.956693518
## 1852 -2.4376451947  0.4665734357 -2.215932535
## 1853  2.9733424483 -0.3397025661 -0.619992205
## 1854 -3.1351828813 -2.5816398580 -1.590792335
## 1855  0.7807817289  0.3814828999  1.901666298
## 1856 -3.2720907928 -0.2049398036 -2.727829282
## 1857  3.9577273459 -1.1379336108 -1.121032887
## 1858  2.4981092965  1.1993207021  0.975247753
## 1859  0.3882511111 -0.0849604987  1.037429554
## 1860  4.1550623247 -0.8675825177  0.725320642
## 1861  4.9150795570 -2.1170065246 -0.593117029
## 1862 -2.8064842947 -0.5978638520  1.064792790
## 1863  0.4287386632  3.0661832401 -0.513377325
## 1864  2.2319296121 -2.5736576869 -0.022523500
## 1865 -0.1769074392 -0.0446190763 -0.045545019
## 1866 -2.7101614693 -1.2727964731  2.152018855
## 1867  3.7643681544 -1.2710182195 -2.297780932
## 1868  1.2183822381  0.2527309653 -1.894095219
## 1869 -3.0697531900  0.8661691201 -1.630985427
## 1870 -3.0440013820 -1.9838019187  0.326011913
## 1871  1.0050044327 -1.8459649115 -1.088364264
## 1872 -3.2529712407 -0.2677802049  0.248536345
## 1873 -1.4586305950 -0.0983055294 -1.757423284
## 1874  2.8750201549 -1.6210774280 -0.023145731
## 1875  2.7493953684  1.2195517310 -0.522783778
## 1876 -3.0854574339 -0.9364589506 -0.102517132
## 1877 -2.6579451881 -1.3604061950 -0.178329293
## 1878 -3.1136375507 -0.6112721789 -0.133349162
## 1879  4.1119173873 -2.6161649285 -0.672548536
## 1880  3.3681174110  0.4202549274  1.264252811
## 1881  4.1212805021 -0.8628994499  0.017047256
## 1882  3.2865226748  1.1701706535  1.287451713
## 1883  0.7836735474  3.9263220105 -1.136482109
## 1884  0.0516814572  2.8991163177  0.468111480
## 1885 -2.4728729850 -1.7515693534  1.157444726
## 1886 -3.1292352366  0.0014754093 -0.830057184
## 1887 -1.2038744781 -1.7197464092  1.554740510
## 1888 -4.4895852933  0.6737511430 -1.159594620
## 1889 -2.2289287084 -2.2443762008  0.437564238
## 1890  3.0647440016 -0.9835001000  1.181535870
## 1891  4.5807004112 -0.6207366070 -0.538393500
## 1892  4.3242298800 -0.0861219002 -1.348310953
## 1893  3.4166375454  1.5968114223  0.223595629
## 1894 -3.7241848331 -1.3243515170 -0.213020605
## 1895  0.0509662855  2.5487011933 -1.346185274
## 1896  2.7032910950 -1.6051203717  0.171813424
## 1897  3.4706476797  0.1634953346  1.124092749
## 1898  1.6058338475  2.5710468173  0.420923444
## 1899 -2.2517069596 -0.0272852325 -1.688433070
## 1900  1.3171611324  0.4439228445 -1.499501512
## 1901 -3.0632625680 -0.6367429085 -0.092355722
## 1902 -2.8143456064  0.0840040234 -0.576939591
## 1903  3.3764085466 -0.0306392464  0.072616397
## 1904 -3.3396043262 -0.2307682500  1.609862932
## 1905  1.2947318968  0.9888499195 -1.163388195
## 1906  0.3752918863  2.7811598439 -1.468731309
## 1907 -4.2472767521 -1.3221922405 -2.032528280
## 1908 -3.0584035858 -1.8517518898  0.527424873
## 1909  3.2994196573 -0.9690170353 -2.482542970
## 1910 -3.2813766268 -0.3209041311 -0.312410913
## 1911 -3.4183444809 -0.6074153330  1.056207021
## 1912  4.0154998649 -2.3072727389  0.576741342
## 1913 -1.3768955815  0.3302558806  0.944354460
## 1914  3.1970342787 -0.6476252209 -0.009657616
## 1915  3.6542305823 -1.4443549542  1.613813284
## 1916  3.1382774056 -1.5917343860 -1.275326153
## 1917 -2.6748575698  0.4734686876  0.578705068
## 1918 -1.4098810496  0.0501085768  0.124980729
## 1919  1.7753203853 -0.7800285375  0.665180485
## 1920  3.6419307793 -1.4874018464 -1.014620995
## 1921  2.7496033074  0.8627846934 -0.558777866
## 1922  0.8671907957 -0.3691499977 -1.678976802
## 1923  3.1579787510 -1.7548361620 -2.745839383
## 1924 -4.0884251944 -1.6148955408 -0.568333360
## 1925  0.7064417606  1.5023379522  1.241612303
## 1926  4.5545473161 -1.6150416387 -0.196651013
## 1927 -1.2598392589  0.4702164947 -2.288382322
## 1928  3.2003677929  0.8242864426  0.336376030
## 1929  1.8931337147  0.7383234425 -2.286445770
## 1930 -3.0319657798 -0.7750635164 -0.880050961
## 1931 -1.3753432663  1.1814377239  0.411003526
## 1932 -2.1292646902 -0.3682146373 -0.810540688
## 1933  4.5341592769 -0.6662962371 -0.198589876
## 1934  2.5519188361  0.3325536545  1.055930550
## 1935  3.6072838999 -0.5551210935 -0.682974159
## 1936  4.3649682963 -1.4945668582 -0.562033928
## 1937  3.3828064783 -1.7597647273 -1.972509548
## 1938  1.3436880825  0.2005020711 -0.196427345
## 1939 -3.2470967570 -1.5929020643 -0.877210317
## 1940 -0.6802897109  1.7535125026  0.800863929
## 1941  0.3191537234  0.3490851335  0.979026342
## 1942 -0.6083383719 -0.1295572705  0.743311433
## 1943  0.0405416493  1.4001380054  1.045623611
## 1944  1.7481421306 -0.8351587012  2.396969848
## 1945 -2.9305541141  1.6860115339 -1.225124192
## 1946 -2.9131900120 -0.8951304210 -0.293852924
## 1947 -3.6022471718 -0.5338691757  0.540835858
## 1948 -1.1758393687  0.1161849345  2.285010197
## 1949 -2.0432118083  0.9752979927  0.449334659
## 1950  5.3104411978 -1.8267962762 -1.256616170
## 1951 -1.0773361363 -0.5045980898 -1.133586533
## 1952 -0.3390636890  2.7637226541  0.932403310
## 1953 -3.8438867784 -0.7304714598 -0.058717800
## 1954 -1.3526934633  2.0858515695  0.586759928
## 1955  0.7969833930 -1.7947896985  1.287754222
## 1956 -1.7321651328 -0.0129419567  1.697923443
## 1957  1.0023189549 -0.6333670367 -1.101963500
## 1958  3.9189670889 -0.5644542145 -1.367887459
## 1959 -0.9539085593 -0.0557035824  0.410235632
## 1960  4.9732094590 -0.0946017603 -1.744139948
## 1961  3.2609783215 -0.7429163483 -0.999633622
## 1962 -4.1954613959  0.1635263592  0.327472988
## 1963  3.9657031599 -2.2402787113 -0.423757120
## 1964  2.8625624423 -0.7933022045  0.904183314
## 1965  1.4154131713  0.6049935386  0.156412087
## 1966 -0.5455817683  2.0766618205 -0.587298272
## 1967  0.9732740370 -1.8659765344  0.947344576
## 1968 -3.6064284057 -1.2619105588  0.248883708
## 1969 -4.4076225927 -0.9207609627 -1.470145263
## 1970 -1.4419271609  0.1383229965  1.978512356
## 1971 -3.9454653050 -2.5787870265 -1.471731065
## 1972 -3.1199580716  1.0756217470 -0.938084275
## 1973 -3.6177020480 -0.1467822659  0.223142953
## 1974  3.3798884295  0.8747000587  1.851543873
## 1975 -0.7032407410  1.5997399838  0.135089779
## 1976  0.3590716832 -0.3623444885 -1.865367800
## 1977  1.6114558928  1.1325107821 -1.473149824
## 1978  3.6252446705 -1.0498662639  0.283013767
## 1979 -1.5859784756  0.1182343689  1.174285146
## 1980  4.9478742757 -1.2271300368 -0.484758169
## 1981  2.1485595223  1.1340967040  3.555327941
## 1982  4.0453048721 -0.5837022285  0.339649855
## 1983 -4.0679414570 -0.9815917885  0.584276772
## 1984 -2.2573320776 -1.6916469056  1.403619094
## 1985 -3.6966846640  0.5371956795  0.368570739
## 1986 -4.6681893615 -0.1438608207 -1.514130828
## 1987 -4.5896885104  0.5509631623  0.651452242
## 1988 -2.1891199587 -0.8980762848  0.534834295
## 1989 -1.9842872996 -0.9861578286  0.313634345
## 1990 -0.4592721285  1.1661774250 -0.123657052
## 1991  1.8027436932  0.5957417339 -2.006107901
## 1992  3.5080524695  1.1270310913 -0.785559576
## 1993 -2.7596843082 -1.1909539027 -0.669303992
## 1994 -4.0586585052 -0.5594121133 -0.230160539
## 1995 -2.5746909688 -1.5557870164 -0.277899442
## 1996  2.2993824243  2.6731962458  0.167960881
## 1997 -2.3560131888 -0.0063780275  3.739093241
## 1998  2.4250421925  1.8891432581 -1.925705100
## 1999  2.4192049865  1.0766662496  0.222710698
## 2000 -3.6182304287  0.0449210830  0.604682407
## 2001  3.3869025957  1.4243079266  1.964156786
## 2002  0.3561694632 -0.1418959569  1.766572504
## 2003  1.2139465571 -0.4473927237 -0.508378256
## 2004  3.1635289523  2.1282442153  1.921475304
## 2005  2.0527780457  0.3536513595 -0.159168676
## 2006 -2.4626274216 -0.9911202405  0.976820745
## 2007  3.8352301160 -0.3224013181  1.483374977
## 2008  2.2957409731 -1.4977262072  1.769973036
## 2009  2.6437306069  3.8551460036  2.106376678
## 2010 -2.9382759370 -1.1859247508  0.822155134
## 2011 -3.1561029804 -0.7224873829 -0.838236251
## 2012  0.8404390314  2.0750058200  2.724400163
## 2013 -0.5570280907  0.0818227391  0.983961335
## 2014 -1.9533382512 -0.5655338729 -1.811020119
## 2015  3.4868059688 -0.5101648409 -0.216288067
## 2016  2.9720696709 -0.2264194591  1.386681017
## 2017  3.7933626082 -1.2981909522 -0.340824668
## 2018 -3.2617861773 -0.3016108674 -1.040934900
## 2019  0.1672995947  2.4670124477  0.790751358
## 2020  2.3215344924 -0.7546469418 -0.581791657
## 2021 -3.5381366346  0.8670057814 -1.858791104
## 2022 -2.9916057161 -0.5352539588 -1.529032244
## 2023 -1.4372201300 -0.9637419096  1.225861538
## 2024  2.7042117305  0.8585133327  0.714333114
## 2025  3.1658438485  0.9052811622 -0.013271522
## 2026  1.7725498070  1.9820013433 -0.244074240
## 2027 -1.6630077335  0.1405420536  0.661940635
## 2028  3.2390065310 -1.1394687250 -0.280414060
## 2029 -3.7351314157 -0.3339873619 -0.666701640
## 2030  3.0963401858  1.7914210562  1.462965973
## 2031  2.4675256396 -0.4206736673  0.028012278
## 2032 -1.1248908459 -1.4613809791 -0.376922822
## 2033  2.2846849426  1.3007559564 -0.881057926
## 2034 -1.8440019464 -1.2178167723  0.698558184
## 2035 -3.0582100179 -0.9317868901  0.598487419
## 2036  0.7356651889  1.0677809199 -0.420218020
## 2037  2.7606037280 -0.5380265396  0.195958561
## 2038 -3.7353747220 -0.5194968460  0.148490087
## 2039  2.2249822436 -0.7394259970  0.487116725
## 2040  3.6339925230 -0.7710389773  1.566864828
## 2041  2.0958909110 -0.7870435950 -0.172917877
## 2042 -4.0373919246 -0.7137701075 -0.608560093
## 2043 -1.7524287175 -2.2990538366  0.867781792
## 2044 -1.1967628766 -0.0950210975  2.333427017
## 2045  0.0605008967  1.8892853777  1.232388848
## 2046  3.8746042999 -0.4807630394  0.878102472
## 2047  2.6131926577  0.8320261836  0.874325247
## 2048  2.9647932664 -1.7124987505  1.742066859
## 2049  3.9815543981 -1.0060256029 -1.011183166
## 2050 -2.4608911646 -1.3097117821  0.240706330
## 2051 -3.6095503030 -1.3090147276 -0.850415445
## 2052 -0.1768499645  2.5314004550 -0.059367190
## 2053 -3.8846584767 -2.0946501112  1.321707858
## 2054  1.0398632459 -1.4754105592 -0.354409543
## 2055  4.6059600898 -1.9302713462  1.376796756
## 2056  2.2993824243  2.6731962458  0.167960881
## 2057 -3.3845458123  0.2572267202 -2.457671992
## 2058  4.1696915335 -2.1818030924  0.796019667
## 2059 -0.7844392457  1.7304389800  0.088646981
## 2060  3.0613815566  1.0957155352 -0.056040305
## 2061  1.6917146059 -0.0218863853  0.667126495
## 2062 -3.3230281078 -0.7888119277 -1.768013150
## 2063  1.8665825190  0.6420421797 -3.237942789
## 2064 -3.1314515516 -0.4718612058  2.477819559
## 2065  2.6961947215  0.2333340877 -1.820429915
## 2066 -2.2103753386 -2.0211764105  1.980071868
## 2067 -2.8708198523 -0.9361309214  2.758636956
## 2068 -1.0773361363 -0.5045980898 -1.133586533
## 2069  1.1810094016  1.0747371960  0.142936003
## 2070 -0.8882242749  1.4612282896 -0.846736577
## 2071  4.2572922024 -0.6825584001  0.035518050
## 2072  1.3618546797  2.4926628495  0.974657214
## 2073  2.2918373642  0.7769201338  0.152327918
## 2074 -3.8317262543 -0.6669234990 -1.408246666
## 2075 -1.4083294215 -0.7586255636  0.031298500
## 2076  0.1980664435  2.1790158091 -0.076243126
## 2077  2.6624948828  0.4618965146 -0.505759722
## 2078 -3.1389859494 -1.4064670506  2.098173806
## 2079  2.9031176515  0.2400879654 -2.637225851
## 2080  4.0408379702 -1.1442230682 -0.505978793
## 2081 -3.5712448251 -0.8679823953  1.040688984
## 2082 -1.6340832196  0.1501648798  0.689802223
## 2083 -4.9407507154 -0.6798981861  0.708531579
## 2084 -2.2657621986  0.8009262844  1.123766851
## 2085  2.3598621942  0.9775996265  2.251488898
## 2086 -3.2526131695 -0.0473904215 -2.006981792
## 2087  4.3920196220 -1.0244969653 -0.421082144
## 2088  0.8762051640  0.8285550249  0.067409608
## 2089  2.9311936525 -0.8350855736 -1.568690419
## 2090  2.1369111736  1.7820791652 -0.692463107
## 2091 -5.3944164651 -0.0304566486 -0.253924661
## 2092 -3.3322778244 -1.0753349790  0.335755918
## 2093  2.4487356516 -0.9178063305 -1.705622024
## 2094 -2.7790581522 -0.5265189314  3.362565207
## 2095  2.5518031739 -0.8094147577 -1.014510722
## 2096 -0.0657409406  0.1438592858 -0.627532687
## 2097 -0.9457925630  1.3340827983  0.295866377
## 2098 -3.8271377451 -1.6126324226  0.076403899
## 2099 -0.3384519833 -1.2708317703 -2.104563788
## 2100 -1.7947591065  2.0554681211  0.741773483
## 2101 -4.2187476983 -1.5590022352  0.033816227
## 2102 -3.3559656836 -0.4242544674 -0.580679629
## 2103 -2.2468476676  0.9189708544  0.450774834
## 2104 -2.0909741871 -1.7899515172  0.070108691
## 2105  1.3412981049  2.1136197208 -0.341427372
## 2106  3.5264386626 -0.0702603205 -0.019814419
## 2107 -4.0116110961 -0.1790518272 -1.669456863
## 2108 -2.9161073653  1.2013208167 -1.479691294
## 2109 -0.1162850550  1.1829292567 -0.944258607
## 2110 -2.7191689899 -0.1937162279 -2.353562152
## 2111 -3.3818301655  0.7067047165 -1.508932218
## 2112  3.4072066995 -0.3869115490  1.382952898
## 2113 -2.5064884971 -0.3612021285  1.236951713
## 2114 -0.9457925630  1.3340827983  0.295866377
## 2115 -3.4342495676 -0.9184879674 -0.058819942
## 2116 -2.6079593656 -1.4531225222  0.614623739
## 2117 -1.2116634573  0.5417300652  0.739901845
## 2118 -3.2611160859 -1.8825101881 -1.381160562
## 2119  1.2470164107  0.9279206369 -2.354764344
## 2120  0.2807293675 -1.3593980587 -0.433171885
## 2121  3.5701208783  1.8681675940  0.195417423
## 2122 -0.1858052043 -0.6282026909 -0.153357183
## 2123  0.6623791364  1.4319496406  0.674866110
## 2124  4.3611727192 -0.9430641356  0.436531744
## 2125 -3.9809990447 -1.5956126339  2.177734520
## 2126 -2.7362033218  0.3636979084 -0.741150225
## 2127  3.5247157418 -0.0861608496  0.350722066
## 2128  2.8451143758 -1.5082114741 -0.997984426
## 2129  4.5899844734 -2.1035976399  0.605116736
## 2130 -2.9444056802 -0.8539024025  0.114845368
## 2131  0.7210673944  0.2635453652 -0.928830914
## 2132  2.8260001231  0.7444412963  0.287217204
## 2133  2.8902941581 -0.8480313767  0.673836234
## 2134  2.6289146531  0.5888451803  0.103250601
## 2135  0.7494188614  1.2573430706 -1.819548770
## 2136  4.5190491949 -0.9644995624 -1.050377087
## 2137  4.0175842422 -0.7676011911  0.434280807
## 2138 -0.2909789169  2.6491336138  0.184694865
## 2139  0.8491698954  0.5674631016  0.983262016
## 2140 -0.4324660132  0.1314697669  0.031537503
## 2141 -3.1166121336  0.2963181770  0.695119855
## 2142 -1.1500469112  2.5082861000 -1.537969192
## 2143 -0.8396558548 -2.5623084667 -0.122931390
## 2144 -2.2662348083 -1.3153401493  1.984971128
## 2145 -3.3413324947 -1.5777610889 -1.506615114
## 2146  3.1056292567  1.1343360552  0.254150071
## 2147  2.5288245167 -0.1470253890 -1.425664537
## 2148  4.2733878410 -0.8949644066 -2.312409147
## 2149  2.5335605858 -0.7455788020 -1.039761135
## 2150 -2.6676372477 -2.1275022762  0.462763817
## 2151  4.4004113717 -1.1891415655 -0.353472716
## 2152 -3.1891774979 -0.2158785772  0.178867912
## 2153 -3.8651115309 -1.1640551994 -0.363254338
## 2154  5.1052803221 -0.6343959179  0.804024963
## 2155  3.5668557775  1.3231393475  0.872018112
## 2156 -3.8736367474  1.4179556425 -0.353717473
## 2157 -1.2806142104 -1.4207454827  0.075762028
## 2158  0.9636916126  0.3760927684  0.433707840
## 2159 -0.1985713593  2.9153096636 -0.299026613
## 2160 -3.4900028735  0.7224561138 -0.252612927
## 2161 -2.2191396779 -1.2591363171  2.875547377
## 2162  2.1557558117  1.3477489725  1.272589705
## 2163 -0.1470825371  2.6379284773  0.045608397
## 2164  3.1860940921 -0.7738781483  0.756127584
## 2165 -2.4728729850 -1.7515693534  1.157444726
## 2166 -2.4797783193  2.7377547579  0.764269848
## 2167  2.3316259632  1.1437132806  2.385811682
## 2168 -3.6213863820  0.9501445875 -1.353709806
## 2169 -3.2189504164 -1.0630816194  1.838165591
## 2170 -3.3459407591 -0.0268111910 -0.845443352
## 2171  0.3485281893  2.1897938102  1.268866025
## 2172  3.5124472934 -1.6517066124 -0.061004172
## 2173 -3.5441245627  0.1662697310 -2.049992854
## 2174  3.4853422475 -1.8200980230  1.932245663
## 2175 -4.5880622878 -0.5710562165  1.313319103
## 2176 -4.2236372715  0.0854627696 -0.063214512
## 2177 -3.0314123755 -0.5360920966  1.008749274
## 2178  3.9873138038 -1.8513414954 -1.967074898
## 2179 -4.7225940819 -0.9626452024  0.200258543
## 2180  0.2691707977  1.3922542308  1.690820211
## 2181  1.1204562040 -0.2943990285  0.738033312
## 2182  3.9284316382 -1.4812804902 -0.890546150
## 2183 -2.9875007297 -0.4701000687  2.281780191
## 2184 -3.2012335538  0.4050796181 -1.523044123
## 2185  0.8654217679  0.8101318167 -2.209580159
## 2186  0.5909890805  3.0335309168  1.477352884
## 2187  0.7637774066  3.5558811704  1.622757106
## 2188  0.3169692393  1.3756157200  0.011940321
## 2189 -3.5404799578 -0.5562583787  0.187346988
## 2190 -2.6682146310  0.0839695260  1.581629015
## 2191  1.9041907564  1.4121932007  0.602560833
## 2192 -3.3564723314 -2.1711185516  1.109076641
## 2193 -3.4454583459 -1.2406639968  0.743889507
## 2194  2.6559170909  0.5366259407  1.175465353
## 2195 -1.3407201927  4.2745295166 -2.960382019
## 2196  2.2098320655 -1.3806840407  0.045170615
## 2197  2.7966772976 -0.0771899288 -1.270260193
## 2198 -1.3785120728  2.0657377203 -1.119772435

Clustering

Elbow/Silhouette methods for the selection of optimal number of clusters

# Elbow method
fviz_nbclust(Customers_clust, hcut, method = "wss") +
geom_vline(xintercept = 4, linetype = 2) +
labs(subtitle = "Elbow method")

# Silhouette method
fviz_nbclust(Customers_clust, hcut, method = "silhouette") + labs(subtitle = "Silhouette method")

Hierarchical Clustering (bottom-up agglomerative approach)

  • Step 1: preparing the data

  • Step 2: computing similarity information between every pair of objects in the data set

  • Step 3: using linkage function to group objects into hierarchical cluster tree

  • Step 4: determining where to cut the dendrogram into clusters

hc.complete = hclust(dist(Customers_clust), method = "complete")
hc_complete = cutree(hc.complete, 3)
table(hc_complete)
## hc_complete
##   1   2   3 
## 996 894 308

Dendrogram plot

fviz_dend(hc.complete, k = 3,  # Cut in three groups
  cex = 0.5,  # Label size
  k_colors = c("blue", "red", "green"),
  color_labels_by_k = TRUE, # Colour labels by groups
  rect = TRUE, # Add rectangle around groups
  ylim = c(3,15)) # Zoom in the dendrogram
## Warning: The `<scale>` argument of `guides()` cannot be `FALSE`. Use "none" instead as
## of ggplot2 3.3.4.
## ℹ The deprecated feature was likely used in the factoextra package.
##   Please report the issue at <https://github.com/kassambara/factoextra/issues>.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

K-Means Clustering

We do the k-means clustering to compare the clustering results we got from hierarchical clustering.

set.seed(123)
km = kmeans(Customers_clust, 3, nstart = 20)
km_clusters = km$cluster
km_clusters
##    [1] 2 1 2 1 3 3 3 1 1 1 1 2 3 1 2 1 3 2 1 3 3 3 3 3 1 3 1 2 1 2 1 3 2 1 3 1 1
##   [38] 2 2 1 1 1 2 1 1 3 2 2 1 2 3 2 2 1 3 2 3 3 3 2 1 1 2 2 3 2 2 3 1 1 2 2 1 3
##   [75] 1 1 1 2 2 1 1 3 2 1 1 1 1 3 1 2 2 1 1 2 2 2 1 1 3 1 2 2 2 3 2 3 1 2 3 1 1
##  [112] 3 1 1 1 2 2 1 3 3 3 2 3 2 1 1 1 1 2 3 2 3 3 3 1 1 3 1 3 3 3 1 3 2 3 1 1 2
##  [149] 1 2 1 2 1 2 1 2 1 1 1 1 1 1 2 2 1 1 2 1 1 2 1 1 1 1 3 2 1 1 2 1 3 1 2 2 2
##  [186] 3 3 2 2 1 1 1 3 1 3 3 2 3 3 2 3 1 2 3 2 1 2 3 1 3 1 3 3 2 1 3 2 1 1 2 1 1
##  [223] 2 1 2 2 1 2 3 1 2 2 2 2 1 1 3 3 1 3 3 1 1 1 3 3 1 3 1 2 3 2 1 2 1 1 1 1 3
##  [260] 2 2 2 3 1 3 3 2 1 1 2 3 2 3 1 1 2 1 1 3 1 1 2 3 1 3 1 3 1 2 1 2 3 1 1 1 2
##  [297] 3 1 1 2 1 2 1 1 3 3 2 1 1 1 1 1 1 3 1 1 2 2 1 2 2 1 3 3 1 2 1 2 1 1 2 2 2
##  [334] 2 3 1 1 2 2 3 2 3 1 1 3 3 2 1 2 3 1 3 1 3 1 1 1 1 2 1 1 1 3 3 1 1 3 2 1 3
##  [371] 2 1 2 3 3 2 3 1 1 1 1 2 3 3 3 1 1 3 1 3 1 3 3 3 2 2 1 2 2 3 1 1 1 2 1 2 2
##  [408] 1 2 2 3 3 2 1 1 2 3 1 1 1 1 1 1 1 1 3 2 1 3 2 2 1 2 3 3 2 3 3 2 3 2 1 2 1
##  [445] 2 2 1 3 3 2 1 3 1 1 3 1 3 3 3 1 1 3 1 2 2 3 3 1 1 2 3 2 3 3 3 1 2 2 2 1 1
##  [482] 1 3 1 2 2 2 1 3 1 2 3 2 3 2 1 1 2 2 1 2 3 2 1 1 2 2 2 3 3 2 3 3 1 3 1 2 3
##  [519] 1 1 3 3 2 2 1 2 1 1 1 1 1 3 1 2 3 2 2 1 2 1 2 2 2 3 3 3 1 1 1 3 1 1 3 1 3
##  [556] 1 1 1 3 1 3 1 1 1 1 2 2 3 1 1 2 2 1 2 1 1 1 1 1 1 3 2 3 1 3 1 1 1 2 3 3 1
##  [593] 1 2 3 1 1 3 3 1 2 1 2 1 2 3 1 1 2 2 2 1 2 1 2 2 3 2 2 2 3 3 2 3 3 2 3 2 1
##  [630] 3 1 3 1 3 2 3 3 1 2 1 1 1 1 1 1 3 2 3 2 2 3 1 3 2 1 2 2 2 3 3 2 2 2 2 2 3
##  [667] 2 3 1 1 1 1 1 2 2 3 2 3 2 3 2 1 3 3 1 1 3 1 2 3 2 2 1 3 1 3 3 1 2 1 1 2 2
##  [704] 3 2 1 3 3 3 2 2 2 2 1 2 3 1 1 1 2 2 1 2 1 2 2 3 2 2 2 2 3 3 1 1 1 3 2 1 2
##  [741] 3 2 2 1 3 2 2 3 1 1 1 1 2 1 2 2 1 1 1 1 1 1 3 3 3 2 2 1 3 1 3 3 3 2 1 3 1
##  [778] 3 2 2 3 1 3 3 2 1 1 2 3 2 3 1 3 3 1 2 1 3 1 2 2 3 1 2 1 1 3 3 1 1 2 3 2 1
##  [815] 3 1 3 1 1 2 2 2 3 3 1 3 3 2 1 3 3 1 3 1 2 1 3 1 3 3 3 3 1 2 1 2 2 1 1 2 2
##  [852] 3 1 2 1 1 1 1 1 2 2 3 1 3 2 1 3 2 1 2 3 2 2 3 1 2 1 2 2 3 2 2 1 1 1 2 2 3
##  [889] 1 2 2 2 2 3 2 1 2 1 1 2 3 2 2 2 2 2 1 3 1 2 1 2 3 2 3 3 3 2 2 1 3 2 3 1 3
##  [926] 1 1 3 1 1 1 3 3 1 3 2 3 1 1 3 2 2 1 1 3 2 1 1 3 2 2 2 3 1 3 1 1 1 2 2 3 2
##  [963] 2 2 1 2 1 3 2 1 1 2 1 2 3 3 2 3 1 3 3 2 2 1 1 2 1 3 1 1 2 2 3 1 1 1 1 3 1
## [1000] 1 2 1 1 1 3 2 2 2 1 2 3 1 1 1 3 2 1 3 2 1 1 1 2 3 3 2 1 2 1 1 2 3 3 2 2 3
## [1037] 3 2 1 3 1 2 3 1 2 1 2 3 1 3 2 2 1 3 1 2 3 2 1 2 3 1 2 1 2 2 1 2 1 1 2 2 2
## [1074] 1 2 2 3 3 3 1 2 1 1 2 3 2 2 1 2 1 3 1 2 1 3 2 1 3 1 3 3 3 1 1 2 3 1 1 2 2
## [1111] 3 1 2 1 3 3 1 1 1 2 1 1 3 3 1 2 2 1 2 3 1 3 2 2 2 3 3 3 2 1 3 1 1 2 2 1 1
## [1148] 2 3 1 3 1 2 1 2 3 1 3 1 1 1 1 3 1 1 2 3 1 1 1 3 1 3 2 2 3 3 3 1 2 3 2 1 1
## [1185] 1 3 2 2 2 3 1 2 1 2 1 1 1 2 1 1 2 3 1 3 1 3 1 1 1 1 3 1 2 1 1 1 1 2 2 3 1
## [1222] 1 1 1 3 2 3 2 2 2 3 3 2 1 2 1 2 2 1 1 2 3 1 1 2 2 2 3 3 1 3 1 1 2 3 2 2 1
## [1259] 3 3 1 3 1 2 2 1 1 1 1 1 3 1 3 2 1 2 1 3 2 3 3 2 2 3 3 2 3 2 3 1 3 1 1 1 2
## [1296] 3 2 3 1 3 1 1 1 2 1 3 2 2 3 3 2 1 1 3 3 3 3 1 1 1 1 2 3 2 2 2 1 3 2 2 1 2
## [1333] 2 1 3 1 3 2 3 2 3 1 3 3 3 1 1 3 3 3 3 1 3 3 3 1 2 1 1 2 3 1 1 1 1 3 1 1 3
## [1370] 3 3 3 1 3 3 3 3 2 3 3 2 2 1 2 3 3 1 2 1 1 1 2 3 1 1 1 1 1 1 3 1 1 2 1 3 1
## [1407] 3 1 1 1 1 1 1 2 2 2 2 3 2 2 3 2 2 1 3 2 3 1 2 2 3 3 1 1 1 2 3 2 1 2 1 1 1
## [1444] 3 2 1 2 1 1 3 3 3 1 1 2 2 2 3 2 3 3 3 2 3 1 2 1 3 2 2 3 1 3 3 2 2 3 3 2 3
## [1481] 2 1 2 2 1 3 3 1 1 1 2 2 1 1 1 3 2 1 2 1 3 3 3 1 1 3 1 2 3 3 1 3 2 1 3 1 1
## [1518] 1 3 1 1 2 2 2 3 1 3 1 1 2 1 3 1 2 1 3 3 2 3 3 2 1 2 1 3 3 1 2 1 3 1 2 2 1
## [1555] 2 1 3 2 3 2 1 2 1 3 1 3 1 3 2 1 3 2 2 1 1 1 2 3 3 3 2 1 2 3 1 2 1 3 3 3 1
## [1592] 1 2 3 3 1 3 2 1 1 2 3 1 1 3 3 3 2 1 1 1 3 3 1 2 1 3 2 2 1 1 2 1 1 1 3 2 3
## [1629] 2 3 3 2 1 1 1 3 1 2 2 3 2 2 2 2 1 3 1 3 1 2 1 1 1 3 1 2 3 2 2 3 2 1 1 1 2
## [1666] 1 2 1 2 3 1 1 3 1 1 3 3 2 3 2 2 1 2 1 3 2 1 3 3 1 2 2 2 3 1 3 1 3 2 1 1 1
## [1703] 2 2 3 3 2 3 1 1 3 1 2 3 2 1 2 2 2 3 3 1 3 1 1 1 1 1 2 2 1 2 2 1 2 1 2 1 1
## [1740] 1 3 2 2 1 3 1 3 1 2 1 1 2 3 3 3 3 1 3 2 1 1 2 1 3 1 3 1 3 2 2 2 1 3 1 3 3
## [1777] 1 2 2 1 3 2 2 2 1 2 2 3 1 2 1 1 2 2 1 3 2 2 1 1 1 3 3 3 1 2 3 3 3 1 2 1 2
## [1814] 3 2 1 1 3 1 2 1 2 2 2 3 1 1 2 3 2 1 2 2 2 3 1 1 3 2 1 2 1 3 1 1 3 2 2 2 2
## [1851] 1 1 2 1 3 1 2 3 3 2 2 1 3 2 3 1 2 3 1 1 2 1 1 2 2 1 1 1 2 2 2 2 3 3 1 1 1
## [1888] 1 1 2 2 2 2 1 3 2 2 3 1 3 1 1 2 1 3 3 1 1 2 1 1 2 1 2 2 2 1 1 2 2 2 3 2 1
## [1925] 3 2 1 2 3 1 3 1 2 2 2 2 2 3 1 3 3 3 3 2 1 1 1 1 1 2 1 3 1 3 2 1 2 2 1 2 2
## [1962] 1 2 2 3 3 2 1 1 1 1 1 1 2 3 3 3 2 1 2 3 2 1 1 1 1 1 1 1 3 3 2 1 1 1 3 1 3
## [1999] 3 1 2 3 2 3 2 1 2 2 3 1 1 3 3 1 2 2 2 1 3 2 1 1 1 2 2 3 1 2 1 3 2 1 3 1 1
## [2036] 3 2 1 2 2 2 1 1 1 3 2 2 2 2 1 1 3 1 2 2 3 1 2 3 2 2 1 2 1 2 1 1 1 3 3 2 3
## [2073] 2 1 1 3 2 1 2 2 1 1 1 1 3 1 2 3 2 3 1 1 2 1 2 3 3 1 1 3 1 1 1 1 3 2 1 1 3
## [2110] 1 1 2 1 3 1 1 3 1 3 3 2 3 3 2 1 1 2 2 2 1 3 2 2 2 3 2 2 3 3 3 1 3 1 1 1 2
## [2147] 2 2 2 1 2 1 1 2 2 1 1 3 3 1 1 3 3 2 1 3 3 1 1 1 3 2 1 2 1 1 1 2 1 3 3 2 1
## [2184] 1 3 3 3 3 1 1 3 1 1 2 3 2 2 3

Compare K-Means and Hierarchical Clustering

table(km_clusters, hc_complete)
##            hc_complete
## km_clusters   1   2   3
##           1  24 832  20
##           2 700   0   0
##           3 272  62 288
hc_clusters = hc_complete

Cluster visualisation

Append the result to our original dataset indicating which cluster each customer belongs to.

Customers_cleaned["kmcluster"] <- km_clusters
Customers_cleaned["hcluster"] <- hc_clusters
Customers_cleaned
## # A tibble: 2,198 × 37
##    Education Income Recency MntWines MntFruits MntMeatProducts MntFishProducts
##        <dbl>  <dbl>   <dbl>    <dbl>     <dbl>           <dbl>           <dbl>
##  1         2  58138      58      635        88             546             172
##  2         2  46344      38       11         1               6               2
##  3         2  71613      26      426        49             127             111
##  4         2  26646      26       11         4              20              10
##  5         4  58293      94      173        43             118              46
##  6         3  62513      16      520        42              98               0
##  7         2  55635      34      235        65             164              50
##  8         4  33454      32       76        10              56               3
##  9         4  30351      19       14         0              24               3
## 10         4   5648      68       28         0               6               1
## # ℹ 2,188 more rows
## # ℹ 30 more variables: MntSweetProducts <dbl>, MntGoldProds <dbl>,
## #   NumDealsPurchases <dbl>, NumWebPurchases <dbl>, NumCatalogPurchases <dbl>,
## #   NumStorePurchases <dbl>, NumWebVisitsMonth <dbl>, AcceptedCmp3 <dbl>,
## #   AcceptedCmp4 <dbl>, AcceptedCmp5 <dbl>, AcceptedCmp1 <dbl>,
## #   AcceptedCmp2 <dbl>, Complain <dbl>, Response <dbl>, Age <dbl>,
## #   AgeCategory <dbl>, NumChildren <dbl>, Spending <dbl>, log_Wines <dbl>, …
PC1 <- Customers_clust[,1]
PC2 <- Customers_clust[,2]
PC3 <- Customers_clust[,3]

# Append the cluster result to our dataset
Customers_clust["hcluster"] <- hc_clusters
cluster3d <- plot_ly(Customers_clust, x = ~Customers_clust$PC1, y = ~Customers_clust$PC2, z = ~Customers_clust$PC3, color = ~as.factor(Customers_cleaned$hcluster), colors = c('#636EFA','#EF553B','#00CC96') ) %>%
  add_markers(size = 12)


cluster3d <- cluster3d %>%
  layout(
    scene = list(
      bgcolor = "#e5ecf6",
      xaxis = list(title = "PC1"),
      yaxis = list(title = "PC2"),
      zaxis = list(title = "PC3")
    )
  )

cluster3d

Results Interpretation

Compare clustering results between K-Means and Hierarchical methods

Hierarchical Clustering Summary statistics

hcresult <- Customers_cleaned %>%
  group_by(hcluster) %>%
  summarise_all(mean) %>%
  t() %>%
  round(2)
hcresult
##                         [,1]     [,2]     [,3]
## hcluster                1.00     2.00     3.00
## Education               2.57     2.35     2.57
## Income              68695.69 32949.33 49606.84
## Recency                49.24    48.38    50.06
## MntWines              538.45    34.30   337.78
## MntFruits              49.72     4.06    15.06
## MntMeatProducts       314.15    18.36   109.38
## MntFishProducts        71.37     5.70    21.64
## MntSweetProducts       50.76     4.19    15.63
## MntGoldProds           66.53    13.13    60.17
## NumDealsPurchases       1.79     1.89     5.32
## NumWebPurchases         5.37     1.91     6.22
## NumCatalogPurchases     4.72     0.44     2.24
## NumStorePurchases       8.22     3.08     5.99
## NumWebVisitsMonth       3.71     6.53     7.20
## AcceptedCmp3            0.07     0.07     0.09
## AcceptedCmp4            0.12     0.01     0.09
## AcceptedCmp5            0.16     0.00     0.00
## AcceptedCmp1            0.13     0.00     0.02
## AcceptedCmp2            0.03     0.00     0.00
## Complain                0.01     0.01     0.00
## Response                0.20     0.08     0.19
## Age                    54.10    49.10    54.14
## AgeCategory             2.28     2.01     2.32
## NumChildren             0.52     1.23     1.54
## Spending             1090.99    79.73   559.65
## log_Wines               6.05     2.87     5.53
## log_Fruits              3.34     1.16     1.86
## log_MeatProducts        5.42     2.62     4.36
## log_FishProducts        3.69     1.40     2.15
## log_SweetProducts       3.34     1.18     1.79
## log_GoldProds           3.80     2.19     3.67
## log_Spending            6.85     4.08     6.13
## Relationship            1.64     1.65     1.65
## YearsJoined             7.95     7.83     8.44
## TotalAcceptedCmp        0.71     0.17     0.40
## kmcluster               2.25     1.14     2.87

K-Means Summary Statistics

kmresult <- Customers_cleaned %>%
  group_by(kmcluster) %>%
  summarise_all(mean) %>%
  t() %>%
  round(2)
kmresult
##                         [,1]     [,2]     [,3]
## kmcluster               1.00     2.00     3.00
## Education               2.37     2.47     2.65
## Income              32699.80 73300.87 53377.64
## Recency                48.41    49.14    49.70
## MntWines               27.60   584.81   381.75
## MntFruits               3.85    62.85    16.76
## MntMeatProducts        16.01   397.85   113.31
## MntFishProducts         5.45    90.97    23.13
## MntSweetProducts        4.00    64.17    17.19
## MntGoldProds           12.21    74.90    53.71
## NumDealsPurchases       1.85     1.40     4.05
## NumWebPurchases         1.79     5.26     5.98
## NumCatalogPurchases     0.40     5.53     2.52
## NumStorePurchases       2.99     8.52     6.76
## NumWebVisitsMonth       6.50     3.05     6.31
## AcceptedCmp3            0.07     0.08     0.08
## AcceptedCmp4            0.01     0.11     0.12
## AcceptedCmp5            0.00     0.21     0.02
## AcceptedCmp1            0.00     0.17     0.03
## AcceptedCmp2            0.00     0.03     0.01
## Complain                0.01     0.01     0.01
## Response                0.08     0.24     0.15
## Age                    48.98    53.08    55.29
## AgeCategory             2.00     2.22     2.37
## NumChildren             1.24     0.32     1.25
## Spending               69.12  1275.55   605.86
## log_Wines               2.79     6.20     5.64
## log_Fruits              1.11     3.76     2.14
## log_MeatProducts        2.56     5.79     4.48
## log_FishProducts        1.36     4.17     2.37
## log_SweetProducts       1.14     3.76     2.09
## log_GoldProds           2.15     3.95     3.57
## log_Spending            4.02     7.08     6.24
## Relationship            1.65     1.63     1.65
## YearsJoined             7.83     7.95     8.19
## TotalAcceptedCmp        0.16     0.83     0.41
## hcluster                2.00     1.00     2.03

From this, we can safely use the clustering result from Hierarchical Clustering.

Number of customers in each cluster

barplot(table(hc_complete), main = "Number of customers in each cluster", xlab = "Clusters", ylab = "Counts")

Education level

Customers_cleaned$Education <- as.factor(Customers_cleaned$Education) # change in categorical variable 
ggplot(Customers_cleaned, aes(x = hcluster, fill = Education)) +
  geom_bar(position = "fill") +
  labs(title = "Bar plot of education by cluster", x = "hcluster", y = "proportion") 

Income level

mean_result <- aggregate(data = Customers_cleaned, Income ~ hcluster, mean)
sd_result <- aggregate(data = Customers_cleaned, Income ~ hcluster, sd)
count_result <- table(Customers_cleaned$hcluster)

# Combine mean, sd, and count into a single table
combined_table <- merge(merge(mean_result, sd_result, by = "hcluster"), as.data.frame(count_result), by.x = "hcluster", by.y = "Var1", all.x = TRUE)

# Rename the columns 
colnames(combined_table) <- c("hcluster", "estimated_income", "sd_Income", "sample_count")

#calculate the standard error
combined_table$standard_error <- combined_table$sd_Income/ sqrt(combined_table$sample_count)

#calculate the t_score using 95% confidence interval 
alpha = 0.05
degrees_of_freedom = combined_table$sample_count - 1
combined_table$t_score = qt(p=alpha/2, df=degrees_of_freedom,lower.tail=F)

#calculate the margin of error
combined_table$margin_error <- combined_table$t_score * combined_table$standard_error

#show the table result
combined_table
##   hcluster estimated_income sd_Income sample_count standard_error  t_score
## 1        1         68695.69  12793.98          996       405.3928 1.962351
## 2        2         32949.33  11696.60          894       391.1928 1.962624
## 3        3         49606.84  11566.32          308       659.0524 1.967721
##   margin_error
## 1     795.5229
## 2     767.7643
## 3    1296.8314
ggplot(combined_table, aes(x = estimated_income, y = reorder(hcluster, estimated_income))) + 
  geom_errorbarh(aes(xmin = estimated_income - margin_error, xmax = estimated_income + margin_error)) + 
  geom_point(size = 3, color = "darkgreen") + 
  theme_minimal(base_size = 12.5) + 
  labs(title = "Mean customer household income", 
       subtitle = "For Each Hierarchial Cluster", 
       x = "Income Estimate", 
       y = "Cluster group")

Num of Children

numchildren <- Customers_cleaned %>%
  group_by(hcluster) %>%
  summarise(across(c("NumChildren"), mean))

numchildren_ct <- numchildren %>% gather(key = numchildren, value = Value, NumChildren)

ggplot(numchildren_ct, aes(x = factor(hcluster), y = Value, fill = numchildren)) +
  geom_bar(stat = "identity", position = "dodge") +
  labs(x = "Cluster Groups", y = "Total", title = "Average no. of Children") +
  theme_minimal()

Spending amount

# Create a scatterplot
spending_plot <- ggplot(Customers_cleaned, aes(x = Spending, y = Income, color = as.factor(hcluster))) +
  geom_point() +
  labs(title = "Income and Spending for each Cluster") +
  theme_minimal()

# Show the plot
print(spending_plot)

Spending patterns

goods <- Customers_cleaned %>%
  select(MntWines, MntFruits, MntMeatProducts, MntFishProducts, MntSweetProducts, MntGoldProds, Spending, hcluster)

sum_spending_by_commodity <- goods %>%
  group_by(hcluster) %>%
  summarise(across(c("MntWines", "MntFruits", "MntMeatProducts", "MntFishProducts", "MntSweetProducts", "MntGoldProds"), sum))

spending_by_cluster <- goods %>%
  group_by(hcluster) %>%
  summarise(across(c("Spending"), sum))

proportion <- sum_spending_by_commodity %>%
  mutate(across(-1, ~./spending_by_cluster$Spending))

library(gt)
library(scales)
## 
## Attaching package: 'scales'
## The following object is masked from 'package:readr':
## 
##     col_factor
spend_table <- proportion %>%
  gt() %>%
  data_color(
    columns = c("MntWines", "MntFruits", "MntMeatProducts", "MntFishProducts", "MntSweetProducts", "MntGoldProds"),
    colors = scales::col_numeric(
      palette = "YlGn",
      domain = NULL
    ) 
  )
## Warning: Since gt v0.9.0, the `colors` argument has been deprecated.
## • Please use the `fn` argument instead.
## This warning is displayed once every 8 hours.
spend_table
hcluster MntWines MntFruits MntMeatProducts MntFishProducts MntSweetProducts MntGoldProds
1 0.4935451 0.04557602 0.2879533 0.06541545 0.04652943 0.06098062
2 0.4301647 0.05089644 0.2302405 0.07149070 0.05249572 0.16471199
3 0.6035493 0.02690692 0.1954494 0.03866637 0.02792217 0.10750586

Campaigns performance

campaigns_plot <- ggplot(Customers_cleaned, aes(x = factor(TotalAcceptedCmp), fill = as.factor(hcluster))) +
  geom_bar(position = "dodge") +
  labs(title = "Count Of Promotion Accepted",
       x = "Number Of Total Accepted Promotions") +
  theme_minimal()

campaigns_plot

Deals purchased

deals_plot <- ggplot(Customers_cleaned, aes(x = NumDealsPurchases, fill = as.factor(hcluster))) +
  geom_bar(position = "dodge") +
  labs(title = "Count of Deals Puchased",
       x = "Number Of Deals Purhcased") +
  theme_minimal()

deals_plot

Sales channels

sum_spending_by_commodity <- Customers_cleaned %>%
  group_by(hcluster) %>%
  summarise(across(c("NumWebPurchases", "NumCatalogPurchases", "NumStorePurchases"), mean))

purchases_ct <- sum_spending_by_commodity %>% gather(key = Purchases, value = Value, NumWebPurchases:NumStorePurchases)

ggplot(purchases_ct, aes(x = factor(hcluster), y = Value, fill = Purchases)) +
  geom_bar(stat = "identity", position = "dodge") +
  labs(x = "Cluster Groups", y = "Average Counts", title = "Average Counts of Purchases by Sales Channels") +
  scale_fill_manual(values = c("NumWebPurchases" = "blue", "NumCatalogPurchases" = "green", "NumStorePurchases" = "red")) +
  theme_minimal()

Supervised classification problem

Classifying the clusters

To maintain the consistency of our analysis, we should use the same variables that we used in the clustering analysis.

Customers_PCA["hcluster"] <- hc_clusters

Customers_PCA <- Customers_PCA %>% 
  mutate( 
    InCluster1 = as.factor(if_else(hcluster == 1, 1, 0)), 
    InCluster2 = as.factor(if_else(hcluster == 2, 1, 0)), 
    InCluster3 = as.factor(if_else(hcluster == 3, 1, 0))) 

Tree for classifying cluster 1

# Fit a tree for cluster 1
tree1.Customers = tree(formula = InCluster1~.-InCluster2-InCluster3-hcluster, data=Customers_PCA) 
plot(tree1.Customers) 
text(tree1.Customers, pretty=1, cex = 0.7) 

cv1.Customers = cv.tree(tree1.Customers, FUN=prune.misclass)
cv1.Customers$size
## [1] 10  9  8  6  3  2  1
cv1.Customers$dev
## [1] 177 177 190 190 231 265 996
# Prune the tree for cluster 1
prune1.Customers = prune.misclass(tree1.Customers, best=6)
plot(prune1.Customers)
text(prune1.Customers, pretty=1, cex = 0.7) 

Tree for classifying cluster 2

# Fit the tree for cluster 2
tree2.Customers = tree(formula = InCluster2~.-InCluster1-InCluster3-hcluster, data=Customers_PCA) 
plot(tree2.Customers) 
text(tree2.Customers, pretty=1, cex = 0.7) 

cv2.Customers = cv.tree(tree2.Customers, FUN=prune.misclass)
cv2.Customers$size
## [1] 5 2 1
cv2.Customers$dev
## [1] 118 118 894
# Prune the tree for cluster 2
prune2.Customers = prune.misclass(tree2.Customers, best=2)
plot(prune2.Customers)
text(prune2.Customers, pretty=1, cex = 0.7) 

Tree for classifying cluster 3

# Fit the tree for cluster 3
tree3.Customers = tree(formula = InCluster3~.-InCluster1-InCluster2-hcluster, data=Customers_PCA) 
plot(tree3.Customers) 
text(tree3.Customers, pretty=1, cex = 0.7) 

cv3.Customers = cv.tree(tree3.Customers, FUN=prune.misclass)
cv3.Customers$size
## [1] 14 10  8  5  4  3  2  1
cv3.Customers$dev
## [1] 126 126 130 137 141 195 229 308
# Prune the tree for cluster 3
prune3.Customers = prune.misclass(tree3.Customers, best=5)
plot(prune3.Customers)
text(prune3.Customers, pretty=1, cex = 0.7) 

Predict the response variable

sum(Customers_cleaned$Response == 1)
## [1] 329
sum(Customers_cleaned$Response == 0)
## [1] 1869
glm.fits = glm(Response ~ Education + Income + NumChildren + Recency + NumDealsPurchases + NumWebPurchases + NumStorePurchases + YearsJoined + Relationship + Age + Spending + TotalAcceptedCmp, data = Customers_cleaned, family = binomial)
summary(glm.fits)
## 
## Call:
## glm(formula = Response ~ Education + Income + NumChildren + Recency + 
##     NumDealsPurchases + NumWebPurchases + NumStorePurchases + 
##     YearsJoined + Relationship + Age + Spending + TotalAcceptedCmp, 
##     family = binomial, data = Customers_cleaned)
## 
## Coefficients:
##                     Estimate Std. Error z value Pr(>|z|)    
## (Intercept)       -1.253e+01  1.708e+00  -7.333 2.25e-13 ***
## Education2         7.579e-01  4.370e-01   1.734 0.082875 .  
## Education3         1.093e+00  4.895e-01   2.232 0.025598 *  
## Education4         1.658e+00  4.721e-01   3.512 0.000444 ***
## Income             6.088e-07  1.159e-05   0.053 0.958093    
## NumChildren       -6.878e-01  2.262e-01  -3.040 0.002363 ** 
## Recency           -3.798e-02  4.534e-03  -8.376  < 2e-16 ***
## NumDealsPurchases  1.917e-01  6.604e-02   2.902 0.003704 ** 
## NumWebPurchases    6.584e-02  4.733e-02   1.391 0.164188    
## NumStorePurchases -2.446e-01  5.250e-02  -4.659 3.18e-06 ***
## YearsJoined        1.525e+00  1.849e-01   8.249  < 2e-16 ***
## Relationship      -1.534e+00  2.339e-01  -6.557 5.49e-11 ***
## Age                9.480e-04  9.792e-03   0.097 0.922872    
## Spending          -4.803e-04  4.020e-04  -1.195 0.232139    
## TotalAcceptedCmp   3.450e+00  2.080e-01  16.584  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 1855.80  on 2197  degrees of freedom
## Residual deviance:  572.21  on 2183  degrees of freedom
## AIC: 602.21
## 
## Number of Fisher Scoring iterations: 7

We now remove the insignificant variables one by one

glm.fits1 = glm(Response ~ Education + NumChildren + Recency + NumDealsPurchases + NumWebPurchases + NumStorePurchases + YearsJoined + Relationship + Age + Spending + TotalAcceptedCmp, data = Customers_cleaned, family = binomial)
summary(glm.fits1)
## 
## Call:
## glm(formula = Response ~ Education + NumChildren + Recency + 
##     NumDealsPurchases + NumWebPurchases + NumStorePurchases + 
##     YearsJoined + Relationship + Age + Spending + TotalAcceptedCmp, 
##     family = binomial, data = Customers_cleaned)
## 
## Coefficients:
##                     Estimate Std. Error z value Pr(>|z|)    
## (Intercept)       -1.250e+01  1.624e+00  -7.696 1.41e-14 ***
## Education2         7.613e-01  4.322e-01   1.762 0.078136 .  
## Education3         1.095e+00  4.866e-01   2.251 0.024392 *  
## Education4         1.661e+00  4.688e-01   3.543 0.000395 ***
## NumChildren       -6.850e-01  2.200e-01  -3.114 0.001845 ** 
## Recency           -3.798e-02  4.534e-03  -8.377  < 2e-16 ***
## NumDealsPurchases  1.906e-01  6.307e-02   3.023 0.002506 ** 
## NumWebPurchases    6.613e-02  4.691e-02   1.410 0.158625    
## NumStorePurchases -2.439e-01  5.071e-02  -4.809 1.52e-06 ***
## YearsJoined        1.523e+00  1.802e-01   8.448  < 2e-16 ***
## Relationship      -1.534e+00  2.338e-01  -6.560 5.37e-11 ***
## Age                1.008e-03  9.724e-03   0.104 0.917423    
## Spending          -4.655e-04  2.872e-04  -1.621 0.104980    
## TotalAcceptedCmp   3.450e+00  2.080e-01  16.591  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 1855.80  on 2197  degrees of freedom
## Residual deviance:  572.22  on 2184  degrees of freedom
## AIC: 600.22
## 
## Number of Fisher Scoring iterations: 7
glm.fits2 = glm(Response ~ Education + NumChildren + Recency + NumDealsPurchases + NumStorePurchases + YearsJoined + Relationship + Age + Spending + TotalAcceptedCmp, data = Customers_cleaned, family = binomial)
summary(glm.fits2)
## 
## Call:
## glm(formula = Response ~ Education + NumChildren + Recency + 
##     NumDealsPurchases + NumStorePurchases + YearsJoined + Relationship + 
##     Age + Spending + TotalAcceptedCmp, family = binomial, data = Customers_cleaned)
## 
## Coefficients:
##                     Estimate Std. Error z value Pr(>|z|)    
## (Intercept)       -1.250e+01  1.610e+00  -7.765 8.15e-15 ***
## Education2         7.443e-01  4.294e-01   1.733 0.083054 .  
## Education3         1.055e+00  4.832e-01   2.183 0.029008 *  
## Education4         1.670e+00  4.658e-01   3.586 0.000336 ***
## NumChildren       -6.846e-01  2.184e-01  -3.135 0.001719 ** 
## Recency           -3.790e-02  4.537e-03  -8.353  < 2e-16 ***
## NumDealsPurchases  2.118e-01  6.035e-02   3.510 0.000449 ***
## NumStorePurchases -2.339e-01  5.009e-02  -4.671 3.00e-06 ***
## YearsJoined        1.527e+00  1.787e-01   8.547  < 2e-16 ***
## Relationship      -1.528e+00  2.330e-01  -6.560 5.39e-11 ***
## Age                2.586e-03  9.658e-03   0.268 0.788881    
## Spending          -3.580e-04  2.739e-04  -1.307 0.191235    
## TotalAcceptedCmp   3.445e+00  2.069e-01  16.652  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 1855.80  on 2197  degrees of freedom
## Residual deviance:  574.11  on 2185  degrees of freedom
## AIC: 600.11
## 
## Number of Fisher Scoring iterations: 7
glm.fits3 = glm(Response ~ Education + NumChildren + Recency + NumDealsPurchases + NumStorePurchases + YearsJoined + Relationship + Spending + TotalAcceptedCmp, data = Customers_cleaned, family = binomial)
summary(glm.fits3)
## 
## Call:
## glm(formula = Response ~ Education + NumChildren + Recency + 
##     NumDealsPurchases + NumStorePurchases + YearsJoined + Relationship + 
##     Spending + TotalAcceptedCmp, family = binomial, data = Customers_cleaned)
## 
## Coefficients:
##                     Estimate Std. Error z value Pr(>|z|)    
## (Intercept)       -1.238e+01  1.545e+00  -8.015 1.10e-15 ***
## Education2         7.572e-01  4.265e-01   1.775 0.075831 .  
## Education3         1.072e+00  4.792e-01   2.236 0.025329 *  
## Education4         1.685e+00  4.623e-01   3.645 0.000267 ***
## NumChildren       -6.749e-01  2.156e-01  -3.131 0.001744 ** 
## Recency           -3.794e-02  4.538e-03  -8.361  < 2e-16 ***
## NumDealsPurchases  2.112e-01  6.038e-02   3.498 0.000468 ***
## NumStorePurchases -2.331e-01  4.997e-02  -4.664 3.11e-06 ***
## YearsJoined        1.526e+00  1.787e-01   8.543  < 2e-16 ***
## Relationship      -1.533e+00  2.324e-01  -6.598 4.15e-11 ***
## Spending          -3.488e-04  2.717e-04  -1.284 0.199204    
## TotalAcceptedCmp   3.444e+00  2.069e-01  16.645  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 1855.80  on 2197  degrees of freedom
## Residual deviance:  574.18  on 2186  degrees of freedom
## AIC: 598.18
## 
## Number of Fisher Scoring iterations: 7
glm.fits4 = glm(Response ~ Education + NumChildren + Recency + NumDealsPurchases + NumStorePurchases + YearsJoined + Relationship + TotalAcceptedCmp, data = Customers_cleaned, family = binomial)
summary(glm.fits4)
## 
## Call:
## glm(formula = Response ~ Education + NumChildren + Recency + 
##     NumDealsPurchases + NumStorePurchases + YearsJoined + Relationship + 
##     TotalAcceptedCmp, family = binomial, data = Customers_cleaned)
## 
## Coefficients:
##                     Estimate Std. Error z value Pr(>|z|)    
## (Intercept)       -12.252265   1.537240  -7.970 1.58e-15 ***
## Education2          0.694876   0.422692   1.644 0.100191    
## Education3          1.010858   0.475816   2.124 0.033630 *  
## Education4          1.630891   0.458252   3.559 0.000372 ***
## NumChildren        -0.567129   0.196436  -2.887 0.003888 ** 
## Recency            -0.038237   0.004516  -8.467  < 2e-16 ***
## NumDealsPurchases   0.208783   0.059274   3.522 0.000428 ***
## NumStorePurchases  -0.266813   0.042675  -6.252 4.05e-10 ***
## YearsJoined         1.509965   0.177595   8.502  < 2e-16 ***
## Relationship       -1.538676   0.231947  -6.634 3.27e-11 ***
## TotalAcceptedCmp    3.369045   0.195940  17.194  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 1855.80  on 2197  degrees of freedom
## Residual deviance:  575.86  on 2187  degrees of freedom
## AIC: 597.86
## 
## Number of Fisher Scoring iterations: 7

Predicting the ‘Reponse’ variable using all the data

glm.probs = predict(glm.fits4, type = "response")
glm.pred = rep(0, 2198)
glm.pred[glm.probs > 0.5] = 1
table(glm.pred, as.factor(Customers_cleaned$Response))
##         
## glm.pred    0    1
##        0 1821   77
##        1   48  252
mean(glm.pred == Customers_cleaned$Response)
## [1] 0.9431301

Our logistic regression model correctly predicts the response variable 94.3% of the time.

We now split into training and testing data

set.seed(123)

# Subset a dataframe with only response = 1 
Customers_cleaned_Response1 <- subset(Customers_cleaned, Response == 1)
Customers_cleaned_Response0 <- subset(Customers_cleaned, Response == 0)
dim(Customers_cleaned_Response1) #329 
## [1] 329  37
# We subset a dataframe with only response = 0 and then random sample into equal size with response = 1
Customers_cleaned_Response0_ind <- sample(1:nrow(Customers_cleaned_Response0), nrow(Customers_cleaned_Response1)) 
Customers_cleaned_Response0 <- Customers_cleaned_Response0[Customers_cleaned_Response0_ind, ]

# Merge 
train <- rbind(Customers_cleaned_Response1, Customers_cleaned_Response0)
train_ind <- sample(1:nrow(train), nrow(train)*0.75) 
train <- Customers_cleaned[train_ind, ]

# Extract the remaining observations as testing data
test <- Customers_cleaned[-train_ind, ]

# Re-fit the logistic regression model 
glm.fits5 = glm(Response ~ Education + NumChildren + Recency + NumDealsPurchases + NumStorePurchases + YearsJoined + Relationship + TotalAcceptedCmp, data = train, family = binomial)

# Result
summary(glm.fits5)
## 
## Call:
## glm(formula = Response ~ Education + NumChildren + Recency + 
##     NumDealsPurchases + NumStorePurchases + YearsJoined + Relationship + 
##     TotalAcceptedCmp, family = binomial, data = train)
## 
## Coefficients:
##                    Estimate Std. Error z value Pr(>|z|)    
## (Intercept)       -14.28008    4.14399  -3.446 0.000569 ***
## Education2          1.48484    1.01416   1.464 0.143163    
## Education3          2.37658    1.21590   1.955 0.050631 .  
## Education4          2.10854    1.21319   1.738 0.082208 .  
## NumChildren        -1.32332    0.61756  -2.143 0.032129 *  
## Recency            -0.04154    0.01341  -3.099 0.001942 ** 
## NumDealsPurchases   0.28165    0.15016   1.876 0.060694 .  
## NumStorePurchases  -0.40243    0.12095  -3.327 0.000877 ***
## YearsJoined         1.76272    0.47385   3.720 0.000199 ***
## Relationship       -1.88713    0.67219  -2.807 0.004994 ** 
## TotalAcceptedCmp    4.42919    0.62431   7.095  1.3e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 384.422  on 492  degrees of freedom
## Residual deviance:  87.701  on 482  degrees of freedom
## AIC: 109.7
## 
## Number of Fisher Scoring iterations: 8
# Prediction
glm.probs2 = predict(glm.fits5, newdata = test, type = "response")
glm.pred2 <- ifelse(glm.probs2 > 0.5, 1, 0)
table(glm.pred2, as.factor(test$Response))
##          
## glm.pred2    0    1
##         0 1397   64
##         1   44  200
# Accuracy 
mean(glm.pred2 == test$Response)
## [1] 0.9366569

Association Rule

Load relevant libraries

library(arules)
## Loading required package: Matrix
## 
## Attaching package: 'Matrix'
## The following objects are masked from 'package:tidyr':
## 
##     expand, pack, unpack
## 
## Attaching package: 'arules'
## The following object is masked from 'package:dplyr':
## 
##     recode
## The following objects are masked from 'package:base':
## 
##     abbreviate, write
library(effects)
## Loading required package: carData
## lattice theme set by effectsTheme()
## See ?effectsTheme for details.
library(arulesViz)
  1. Web sales channel
# Extract the relevant features
CustomersAssoc1 <- Customers_cleaned[c("AgeCategory", "NumChildren", "Education", "Relationship", "Recency", "YearsJoined", "NumWebPurchases")]

# Convert into categorical variables using quantiles
CustomersAssoc1$NumWebPurchases <- cut(
  CustomersAssoc1$NumWebPurchases,
  breaks = quantile(CustomersAssoc1$NumWebPurchases, c(0, 0.33, 0.66, 1)),
  labels = c("Low", "Medium", "High"),
  include.lowest = TRUE
)

CustomersAssoc1$Recency <- cut(
  CustomersAssoc1$Recency,
  breaks = quantile(CustomersAssoc1$Recency, c(0, 0.5, 1)),
  labels = c("Recent", "Distant"),
  include.lowest = TRUE
)

CustomersAssoc1$YearsJoined <- cut(
  CustomersAssoc1$YearsJoined,
  breaks = quantile(CustomersAssoc1$YearsJoined, c(0, 0.5, 1)),
  labels = c("<8 years", ">8 years"),
  include.lowest = TRUE
)

CustomersAssoc1$NumChildren <- ifelse(CustomersAssoc1$NumChildren == 0,"NoChild", "YesChild")

CustomersAssoc1$AgeCategory <- ifelse(CustomersAssoc1$AgeCategory == 1,"Millenials", 
                                  ifelse(CustomersAssoc1$AgeCategory == 2, "GenX", "BabyBoomer"))

CustomersAssoc1$Relationship <- ifelse(CustomersAssoc1$Relationship == 1,"Not-Partnered", "Partnered")

CustomersAssoc1$Education <- ifelse(CustomersAssoc1$Education == 1,"Bachelors",
                                    ifelse(CustomersAssoc1$Education == 2, "Graduates",
                                           ifelse(CustomersAssoc1$Education ==3, "Masters", "PhDs")))

# Association rules for web channel
rules_web <- apriori(CustomersAssoc1, parameter = list(support = 0.05, confidence = 0.3), appearance = list(rhs = "NumWebPurchases=High"))
## Warning: Column(s) 1, 2, 3, 4 not logical or factor. Applying default
## discretization (see '? discretizeDF').
## Apriori
## 
## Parameter specification:
##  confidence minval smax arem  aval originalSupport maxtime support minlen
##         0.3    0.1    1 none FALSE            TRUE       5    0.05      1
##  maxlen target  ext
##      10  rules TRUE
## 
## Algorithmic control:
##  filter tree heap memopt load sort verbose
##     0.1 TRUE TRUE  FALSE TRUE    2    TRUE
## 
## Absolute minimum support count: 109 
## 
## set item appearances ...[1 item(s)] done [0.00s].
## set transactions ...[18 item(s), 2198 transaction(s)] done [0.00s].
## sorting and recoding items ... [18 item(s)] done [0.00s].
## creating transaction tree ... done [0.00s].
## checking subsets of size 1 2 3 4 5 6 done [0.00s].
## writing ... [17 rule(s)] done [0.00s].
## creating S4 object  ... done [0.00s].
inspect(rules_web)
##      lhs                          rhs                       support confidence  coverage     lift count
## [1]  {Education=PhDs}          => {NumWebPurchases=High} 0.07051865  0.3283898 0.2147407 1.171755   155
## [2]  {YearsJoined=>8 years}    => {NumWebPurchases=High} 0.08007279  0.3606557 0.2220200 1.286885   176
## [3]  {AgeCategory=BabyBoomer}  => {NumWebPurchases=High} 0.11692448  0.3346354 0.3494086 1.194040   257
## [4]  {NumChildren=YesChild,                                                                            
##       Education=PhDs}          => {NumWebPurchases=High} 0.05186533  0.3294798 0.1574158 1.175644   114
## [5]  {Education=PhDs,                                                                                  
##       YearsJoined=<8 years}    => {NumWebPurchases=High} 0.05050045  0.3016304 0.1674249 1.076272   111
## [6]  {Relationship=Partnered,                                                                          
##       YearsJoined=>8 years}    => {NumWebPurchases=High} 0.05277525  0.3741935 0.1410373 1.335191   116
## [7]  {NumChildren=YesChild,                                                                            
##       YearsJoined=>8 years}    => {NumWebPurchases=High} 0.05914468  0.3757225 0.1574158 1.340646   130
## [8]  {AgeCategory=BabyBoomer,                                                                          
##       Recency=Distant}         => {NumWebPurchases=High} 0.05914468  0.3194103 0.1851683 1.139714   130
## [9]  {AgeCategory=BabyBoomer,                                                                          
##       Recency=Recent}          => {NumWebPurchases=High} 0.05777980  0.3518006 0.1642402 1.255288   127
## [10] {AgeCategory=BabyBoomer,                                                                          
##       Education=Graduates}     => {NumWebPurchases=High} 0.05095541  0.3043478 0.1674249 1.085968   112
## [11] {AgeCategory=BabyBoomer,                                                                          
##       Relationship=Partnered}  => {NumWebPurchases=High} 0.07142857  0.3230453 0.2211101 1.152684   157
## [12] {AgeCategory=BabyBoomer,                                                                          
##       NumChildren=YesChild}    => {NumWebPurchases=High} 0.08553230  0.3686275 0.2320291 1.315330   188
## [13] {AgeCategory=BabyBoomer,                                                                          
##       YearsJoined=<8 years}    => {NumWebPurchases=High} 0.08644222  0.3130148 0.2761601 1.116894   190
## [14] {AgeCategory=BabyBoomer,                                                                          
##       NumChildren=YesChild,                                                                            
##       Relationship=Partnered}  => {NumWebPurchases=High} 0.05641492  0.3746224 0.1505914 1.336721   124
## [15] {AgeCategory=BabyBoomer,                                                                          
##       Relationship=Partnered,                                                                          
##       YearsJoined=<8 years}    => {NumWebPurchases=High} 0.05368517  0.3072917 0.1747043 1.096473   118
## [16] {AgeCategory=BabyBoomer,                                                                          
##       NumChildren=YesChild,                                                                            
##       YearsJoined=<8 years}    => {NumWebPurchases=High} 0.06278435  0.3424318 0.1833485 1.221859   138
## [17] {NumChildren=YesChild,                                                                            
##       Relationship=Partnered,                                                                          
##       Recency=Distant}         => {NumWebPurchases=High} 0.07370337  0.3079848 0.2393085 1.098946   162
  1. Store sales channel
# Subset the relevant variables for the second association rule
CustomersAssoc2 <- subset(CustomersAssoc1, select = -NumWebPurchases)
CustomersAssoc2$NumStorePurchases <- Customers_cleaned$NumStorePurchases

CustomersAssoc2$NumStorePurchases <- cut(
  CustomersAssoc2$NumStorePurchases,
  breaks = quantile(CustomersAssoc2$NumStorePurchases, c(0, 0.33, 0.66, 1)),
  labels = c("Low", "Medium", "High"),
  include.lowest = TRUE
)
rules_store <- apriori(CustomersAssoc2, parameter = list(support = 0.05, confidence = 0.4), appearance = list(rhs = "NumStorePurchases=High"))
## Warning: Column(s) 1, 2, 3, 4 not logical or factor. Applying default
## discretization (see '? discretizeDF').
## Apriori
## 
## Parameter specification:
##  confidence minval smax arem  aval originalSupport maxtime support minlen
##         0.4    0.1    1 none FALSE            TRUE       5    0.05      1
##  maxlen target  ext
##      10  rules TRUE
## 
## Algorithmic control:
##  filter tree heap memopt load sort verbose
##     0.1 TRUE TRUE  FALSE TRUE    2    TRUE
## 
## Absolute minimum support count: 109 
## 
## set item appearances ...[1 item(s)] done [0.00s].
## set transactions ...[18 item(s), 2198 transaction(s)] done [0.00s].
## sorting and recoding items ... [18 item(s)] done [0.00s].
## creating transaction tree ... done [0.00s].
## checking subsets of size 1 2 3 4 5 6 done [0.00s].
## writing ... [9 rule(s)] done [0.00s].
## creating S4 object  ... done [0.00s].
inspect(rules_store)
##     lhs                             rhs                         support confidence  coverage     lift count
## [1] {NumChildren=NoChild}        => {NumStorePurchases=High} 0.12602366  0.4460548 0.2825296 1.531919   277
## [2] {AgeCategory=BabyBoomer,                                                                               
##      NumChildren=NoChild}        => {NumStorePurchases=High} 0.05141037  0.4379845 0.1173794 1.504203   113
## [3] {NumChildren=NoChild,                                                                                  
##      Relationship=Not-Partnered} => {NumStorePurchases=High} 0.05414013  0.4817814 0.1123749 1.654618   119
## [4] {NumChildren=NoChild,                                                                                  
##      Recency=Distant}            => {NumStorePurchases=High} 0.06414923  0.4563107 0.1405823 1.567142   141
## [5] {NumChildren=NoChild,                                                                                  
##      Recency=Recent}             => {NumStorePurchases=High} 0.06187443  0.4358974 0.1419472 1.497035   136
## [6] {NumChildren=NoChild,                                                                                  
##      Education=Graduates}        => {NumStorePurchases=High} 0.06187443  0.4317460 0.1433121 1.482778   136
## [7] {NumChildren=NoChild,                                                                                  
##      Relationship=Partnered}     => {NumStorePurchases=High} 0.07188353  0.4224599 0.1701547 1.450886   158
## [8] {NumChildren=NoChild,                                                                                  
##      YearsJoined=<8 years}       => {NumStorePurchases=High} 0.09463148  0.4342380 0.2179254 1.491336   208
## [9] {NumChildren=NoChild,                                                                                  
##      Relationship=Partnered,                                                                               
##      YearsJoined=<8 years}       => {NumStorePurchases=High} 0.05595996  0.4226804 0.1323931 1.451643   123
  1. Catalog sales channel
CustomersAssoc3 <- subset(CustomersAssoc1, select = -NumWebPurchases)
CustomersAssoc3$NumCatalogPurchases <- Customers_cleaned$NumCatalogPurchases

CustomersAssoc3$NumCatalogPurchases <- cut(
  CustomersAssoc3$NumCatalogPurchases,
  breaks = quantile(CustomersAssoc3$NumCatalogPurchases, c(0, 0.33, 0.66, 1)),
  labels = c("Low", "Medium", "High"),
  include.lowest = TRUE
)

rules_catalog <- apriori(CustomersAssoc3, parameter = list(support = 0.05, confidence = 0.6), appearance = list(rhs = "NumCatalogPurchases=High"))
## Warning: Column(s) 1, 2, 3, 4 not logical or factor. Applying default
## discretization (see '? discretizeDF').
## Apriori
## 
## Parameter specification:
##  confidence minval smax arem  aval originalSupport maxtime support minlen
##         0.6    0.1    1 none FALSE            TRUE       5    0.05      1
##  maxlen target  ext
##      10  rules TRUE
## 
## Algorithmic control:
##  filter tree heap memopt load sort verbose
##     0.1 TRUE TRUE  FALSE TRUE    2    TRUE
## 
## Absolute minimum support count: 109 
## 
## set item appearances ...[1 item(s)] done [0.00s].
## set transactions ...[18 item(s), 2198 transaction(s)] done [0.00s].
## sorting and recoding items ... [18 item(s)] done [0.00s].
## creating transaction tree ... done [0.00s].
## checking subsets of size 1 2 3 4 5 6 done [0.00s].
## writing ... [19 rule(s)] done [0.00s].
## creating S4 object  ... done [0.00s].
inspect(rules_catalog)
##      lhs                              rhs                           support confidence   coverage     lift count
## [1]  {NumChildren=NoChild}         => {NumCatalogPurchases=High} 0.18471338  0.6537842 0.28252957 2.103979   406
## [2]  {AgeCategory=BabyBoomer,                                                                                   
##       NumChildren=NoChild}         => {NumCatalogPurchases=High} 0.08052775  0.6860465 0.11737944 2.207804   177
## [3]  {NumChildren=NoChild,                                                                                      
##       Relationship=Not-Partnered}  => {NumCatalogPurchases=High} 0.07233849  0.6437247 0.11237489 2.071606   159
## [4]  {AgeCategory=GenX,                                                                                         
##       NumChildren=NoChild}         => {NumCatalogPurchases=High} 0.05823476  0.6701571 0.08689718 2.156669   128
## [5]  {NumChildren=NoChild,                                                                                      
##       Recency=Distant}             => {NumCatalogPurchases=High} 0.09508644  0.6763754 0.14058235 2.176681   209
## [6]  {NumChildren=NoChild,                                                                                      
##       Recency=Recent}              => {NumCatalogPurchases=High} 0.08962693  0.6314103 0.14194722 2.031976   197
## [7]  {NumChildren=NoChild,                                                                                      
##       Education=Graduates}         => {NumCatalogPurchases=High} 0.09554140  0.6666667 0.14331210 2.145437   210
## [8]  {NumChildren=NoChild,                                                                                      
##       Relationship=Partnered}      => {NumCatalogPurchases=High} 0.11237489  0.6604278 0.17015469 2.125359   247
## [9]  {NumChildren=NoChild,                                                                                      
##       YearsJoined=<8 years}        => {NumCatalogPurchases=High} 0.14285714  0.6555324 0.21792539 2.109605   314
## [10] {AgeCategory=BabyBoomer,                                                                                   
##       NumChildren=NoChild,                                                                                      
##       YearsJoined=<8 years}        => {NumCatalogPurchases=High} 0.06505914  0.7009804 0.09281165 2.255864   143
## [11] {NumChildren=NoChild,                                                                                      
##       Relationship=Not-Partnered,                                                                               
##       YearsJoined=<8 years}        => {NumCatalogPurchases=High} 0.05414013  0.6329787 0.08553230 2.037024   119
## [12] {NumChildren=NoChild,                                                                                      
##       Education=Graduates,                                                                                      
##       Recency=Distant}             => {NumCatalogPurchases=High} 0.05095541  0.6913580 0.07370337 2.224897   112
## [13] {NumChildren=NoChild,                                                                                      
##       Relationship=Partnered,                                                                                   
##       Recency=Distant}             => {NumCatalogPurchases=High} 0.05505005  0.6875000 0.08007279 2.212482   121
## [14] {NumChildren=NoChild,                                                                                      
##       Recency=Distant,                                                                                          
##       YearsJoined=<8 years}        => {NumCatalogPurchases=High} 0.07324841  0.6880342 0.10646042 2.214201   161
## [15] {NumChildren=NoChild,                                                                                      
##       Relationship=Partnered,                                                                                   
##       Recency=Recent}              => {NumCatalogPurchases=High} 0.05732484  0.6363636 0.09008189 2.047917   126
## [16] {NumChildren=NoChild,                                                                                      
##       Recency=Recent,                                                                                           
##       YearsJoined=<8 years}        => {NumCatalogPurchases=High} 0.06960874  0.6244898 0.11146497 2.009705   153
## [17] {NumChildren=NoChild,                                                                                      
##       Education=Graduates,                                                                                      
##       Relationship=Partnered}      => {NumCatalogPurchases=High} 0.05868972  0.6861702 0.08553230 2.208202   129
## [18] {NumChildren=NoChild,                                                                                      
##       Education=Graduates,                                                                                      
##       YearsJoined=<8 years}        => {NumCatalogPurchases=High} 0.07734304  0.6772908 0.11419472 2.179627   170
## [19] {NumChildren=NoChild,                                                                                      
##       Relationship=Partnered,                                                                                   
##       YearsJoined=<8 years}        => {NumCatalogPurchases=High} 0.08871702  0.6701031 0.13239308 2.156496   195